"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", blockquote: "blockquote", code: "code", details: "details", div: "div", h2: "h2", h4: "h4", img: "img", li: "li", math: "math", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", msup: "msup", mtext: "mtext", ol: "ol", p: "p", path: "path", pre: "pre", semantics: "semantics", span: "span", strong: "strong", summary: "summary", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.details, { "data-callout": true, "data-callout-type": "info", children: ["\n", _jsx(_components.summary, { "data-callout-title": true, children: "转载申明" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "带你发明 Floyd 算法：从记忆化搜索到递推（Python/Java/C++/Go/JS/Rust）" }), "\n", _jsx(_components.a, { href: "https://leetcode.cn/u/endlesscheng/", target: "_blank", rel: "noopener noreferrer", children: _jsx(_components.img, { src: "https://assets.leetcode.cn/aliyun-lc-upload/users/endlesscheng/avatar_1690721039.png", alt: "", className: "internal" }) }), "\n", _jsx(_components.p, { children: _jsx(_components.a, { href: "https://leetcode.cn/u/endlesscheng/", target: "_blank", rel: "noopener noreferrer", children: "灵茶山艾府" }) }), "\n", _jsx(_components.img, { src: "https://pic.leetcode.cn/1672297038-gOvOPc-%E6%8E%A5%E5%8F%97%E6%84%9F%E6%81%A9sm.png", alt: "2022 感恩勋章", className: "internal" }), "\n", _jsx(_components.p, { children: "27706" }), "\n", _jsx(_components.p, { children: "2023.11.14" }), "\n", _jsx(_components.p, { children: "发布于浙江" }), "\n", _jsx(_components.p, { children: "记忆化搜索" }), "\n", _jsx(_components.p, { children: "动态规划" }), "\n", _jsx(_components.p, { children: "最短路" }), "\n", _jsx(_components.p, { children: "C++" }), "\n", _jsx(_components.p, { children: "6+" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["王道书上 Floyd 算法是空间优化版本，对应的，可以直接跳到 ", _jsx(_components.code, { className: "language-auto", children: "空间优化" }), " 部分。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "题意解读", children: [_jsx(_components.a, { className: "heading-link", href: "#题意解读", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "题意解读"] }), "\n", _jsxs(_components.p, { children: ["对于城市 i，求出 i 到其余城市的最短路长度，统计有多少个最短路长度 ≤", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "h" }), _jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "h" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "l" }), _jsx(_components.mi, { children: "d" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "distanceThreshold" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "an" }), _jsx(_components.span, { className: "mord mathnormal", children: "ce" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "T" }), _jsx(_components.span, { className: "mord mathnormal", children: "h" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "r" }), _jsx(_components.span, { className: "mord mathnormal", children: "es" }), _jsx(_components.span, { className: "mord mathnormal", children: "h" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0197em" }, children: "l" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" })] }) })] }), "，把个数记作 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "cnti" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["例如示例 1，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mtext, { children: "" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mtext, { children: "" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mtext, { children: "" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mtext, { children: "" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "cnt0=2,cnt1=3,cnt2=3,cnt3=2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] })] })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["其中 cnti 最小的 i 就是答案。示例 1 中的城市 0 和城市 3 对应的 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "cnti" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 都是最小的，这种情况返回 0 和 3 的最大值，即 3。"] }), "\n", _jsxs(_components.p, { children: ["为了解决本题，我们需要求出", _jsx(_components.strong, { children: "任意两个城市之间的最短路长度" }), "。"] }), "\n", _jsxs(_components.h2, { id: "前置知识", children: [_jsx(_components.a, { className: "heading-link", href: "#前置知识", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "前置知识"] }), "\n", _jsxs(_components.p, { children: ["请看视频讲解 ", _jsx(_components.a, { href: "https://leetcode.cn/link/?target=https%3A%2F%2Fwww.bilibili.com%2Fvideo%2FBV1Xj411K7oF%2F", target: "_blank", rel: "noopener noreferrer", children: "动态规划入门：从记忆化搜索到递推" }), "，制作不易，欢迎点赞！"] }), "\n", _jsxs(_components.h2, { id: "一启发思考寻找子问题", children: [_jsx(_components.a, { className: "heading-link", href: "#一启发思考寻找子问题", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "一、启发思考：寻找子问题"] }), "\n", _jsx(_components.img, { src: "https://pic.leetcode.cn/1699881511-DbyZrc-lc1334-c.png", alt: "lc1334-c.png", className: "internal" }), "\n", _jsxs(_components.h2, { id: "二递归怎么写状态定义与状态转移方程", children: [_jsx(_components.a, { className: "heading-link", href: "#二递归怎么写状态定义与状态转移方程", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "二、递归怎么写：状态定义与状态转移方程"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "定义" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(k,i,j)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 表示从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的最短路长度，并且这条最短路的中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] })] })] }), "。注意中间节点不包含 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), "。"] }), "\n", _jsx(_components.p, { children: "根据上面的讨论：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "不选" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "，那么中间节点的编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\leq k−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", 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className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "选" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "，问题分解成从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 的最短路，以及从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的最短路。由于这两条最短路的中间节点都不包含 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "，所以中间节点的编号都", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤k−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "，故得到", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { 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_jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 之间没有任何中间节点，此时最短路长度只能是连接 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) 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style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的边的边权，即 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "w[i][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "。如果没有连接 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的边，则 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "w[i][j]=∞" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord", children: "∞" })] })] })] }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "递归入口" }), "：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(n−1,i,j)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "，表示从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的最短路长度。", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k=n−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 是因为本题节点编号为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "0" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "，任意最短路的中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤n−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "答疑", children: [_jsx(_components.a, { className: "heading-link", href: "#答疑", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "答疑"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "问" }), "：为什么要强调「中间节点不包含 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), "」？我看代码中没有判断这个呀？"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "答" }), "：只要最短路是简单路径（没有重复节点），中间节点就必然不会包含 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), "。强调这个事实，便于大家理解后文空间优化写法的正确性。"] }), "\n", _jsx(_components.pre, { className: "language-py", children: _jsxs(_components.code, { className: "language-py code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 会超时的递归代码" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "class" }), " ", _jsx(_components.span, { className: "token class-name", children: "Solution" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "findTheCity" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "self", _jsx(_components.span, { className: "token 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className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token builtin", children: "min" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "k ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " i", _jsx(_components.span, { className: "token punctuation", children: "," }), " j", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "k ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", 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className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "复杂度分析"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["时间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "3" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{2}3^{n})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。外面调用 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{2})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 次 ", _jsx(_components.span, { className: "katex-error", title: "ParseError: KaTeX parse error: Can't use function '$' in math mode at position 4: dfs$̲$，", style: { color: "#cc0000" }, children: "dfs$$，" }), " 它的搜索树是一棵满三叉树（每个非叶节点都有三个子节点），树高为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "，所以节点个数为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "3" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(3^{n})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "，遍历搜索树需要 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "3" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(3^{n})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 的时间。"] }), "\n", _jsxs(_components.li, { children: ["空间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。递归需要 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 的栈空间。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "三递归--记录返回值--记忆化搜索", children: [_jsx(_components.a, { className: "heading-link", href: "#三递归--记录返回值--记忆化搜索", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "三、递归 + 记录返回值 = 记忆化搜索"] }), "\n", _jsx(_components.p, { children: "上面的做法太慢了，怎么优化呢？" }), "\n", _jsxs(_components.p, { children: ["比如，从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "2" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "1" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) })] }), " 的最短路可以分解出从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "2" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "7" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "7" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "7" })] }) })] }), " 的最短路（中间节点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mn, { children: "6" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤6" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "6" })] })] })] }), "），从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "2" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "3" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "3" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3" })] }) })] }), " 的最短路也可以分解出从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "2" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "7" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "7" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "7" })] }) })] }), " 的最短路（中间节点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mn, { children: "6" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤6" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "6" })] })] })] }), "）。也就是说，都会递归到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "6" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(6,2,7)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "6" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["一叶知秋，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" })] }) })] }), " 中有大量重复递归调用（递归入参相同）。由于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" })] }) })] }), " 没有副作用，同样的入参无论计算多少次，算出来的结果都是一样的，因此可以用", _jsx(_components.strong, { children: "记忆化搜索" }), "来优化："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["如果一个状态（递归入参）是第一次遇到，那么可以在返回前，把状态及其结果记到一个 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 数组中。"] }), "\n", _jsxs(_components.li, { children: ["如果一个状态不是第一次遇到（", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 中保存的结果不等于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 的初始值），那么可以直接返回 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 中保存的结果。"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["注意：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 数组的初始值一定不能等于要记忆化的值。本题可以初始化成 0，因为边权都是正数。"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["Python 用户可以无视上面这段，直接用 ", _jsx(_components.code, { className: "language-auto", children: "@cache" }), " 装饰器。"] }), "\n"] }), "\n", _jsx(_components.pre, { className: "language-py", children: _jsxs(_components.code, { className: "language-py code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "class" }), " ", _jsx(_components.span, { className: "token class-name", children: "Solution" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "findTheCity" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "self", _jsx(_components.span, { className: "token punctuation", children: "," }), " n", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " edges", _jsx(_components.span, { className: "token punctuation", children: ":" }), " List", _jsx(_components.span, { className: "token punctuation", children: "[" }), "List", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " distanceThreshold", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" w ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "inf", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " n ", _jsx(_components.span, { className: "token keyword", children: "for" }), " _ ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " x", _jsx(_components.span, { className: "token punctuation", children: "," }), " y", _jsx(_components.span, { className: "token punctuation", children: "," }), " wt ", _jsx(_components.span, { className: "token keyword", children: "in" }), " edges", _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" w", _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "y", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " w", _jsx(_components.span, { className: "token punctuation", children: "[" }), "y", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " wt\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token decorator annotation punctuation", children: "@cache" }), " ", _jsx(_components.span, { className: "token comment", children: "# 缓存装饰器，避免重复计算 dfs 的结果" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "dfs" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "k", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " i", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " j", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " k ", _jsx(_components.span, { className: "token operator", children: "<" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token comment", children: "# 递归边界" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " w", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "j", _jsx(_components.span, { className: "token punctuation", children: "]" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token builtin", children: "min" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "k ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " i", _jsx(_components.span, { className: "token punctuation", children: "," }), " j", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "k ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " i", _jsx(_components.span, { className: "token punctuation", children: "," }), " k", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "k ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " k", _jsx(_components.span, { className: "token punctuation", children: "," }), " j", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" ans ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" min_cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " inf\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " i ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " j ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " j ", _jsx(_components.span, { className: "token operator", children: "!=" }), " i ", _jsx(_components.span, { className: "token keyword", children: "and" }), " dfs", _jsx(_components.span, { className: "token punctuation", children: "(" }), "n ", _jsx(_components.span, { className: "token operator", children: "-" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " i", _jsx(_components.span, { className: "token punctuation", children: "," }), " j", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "<=" }), " distanceThreshold", _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" cnt ", _jsx(_components.span, { className: "token operator", children: "+=" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " cnt ", _jsx(_components.span, { className: "token operator", children: "<=" }), " min_cnt", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token comment", children: "# 相等时取最大的 i" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" min_cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " cnt\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ans ", _jsx(_components.span, { className: "token operator", children: "=" }), " i\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ans\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "复杂度分析-1", children: [_jsx(_components.a, { className: "heading-link", href: "#复杂度分析-1", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "复杂度分析"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["时间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。由于每个状态只会计算一次，动态规划的时间复杂度 = 状态个数 × 单个状态的计算时间。本题中状态个数等于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "，单个状态的计算时间为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(1)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "，所以动态规划的时间复杂度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["空间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。有多少个状态，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "o" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "memo" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" })] }) })] }), " 数组的大小就是多少。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "四11-翻译成递推", children: [_jsx(_components.a, { className: "heading-link", href: "#四11-翻译成递推", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "四、1:1 翻译成递推"] }), "\n", _jsx(_components.p, { children: "我们可以去掉递归中的「递」，只保留「归」的部分，即自底向上计算。" }), "\n", _jsx(_components.p, { children: "做法：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" })] }) })] }), " 改成 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "f" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" })] }) })] }), " 数组；"] }), "\n", _jsx(_components.li, { children: "递归改成循环（每个参数都对应一层循环）；" }), "\n", _jsx(_components.li, { children: "递归边界改成 f 数组的初始值。" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["相当于之前是用递归去计算每个状态，现在是", _jsx(_components.strong, { children: "枚举" }), "并计算每个状态。"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["具体来说，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 的定义和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(k,i,j)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 的定义是一样的，都表示从 i 到 j 的最短路长度，并且这条最短路的中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] })] })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["相应的递推式（状态转移方程）也和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" })] }) })] }), " 是一样的："] }), "\n", _jsx(_components.p, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][j]=min(f[k−1][i][j],f[k−1][i][k]+f[k−1][k][j])" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "min" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), 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children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "])" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["但是，这种定义方式", _jsx(_components.strong, { children: "没有状态能表示递归边界" }), "，即 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k=−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 的情况。"] }), "\n", _jsxs(_components.p, { children: ["解决办法：把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "f" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" })] }) })] }), " 数组的长度加一（在前面插入一个二维数组），用 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[0][i][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 表示 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "w" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(−1,i,j)=w[i][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "。由于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[0]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 被占用，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "f" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" })] }) })] }), " 第一维度的下标需要全部向右偏移一位，也就是把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 改为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "，把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k−1]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), " 改为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["修改后", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][i][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), " 表示从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的最短路长度，并且这条最短路的中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" })] }), 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"mclose", children: "]" })] }) })] }), "，翻译自 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "dfs(n−1,i,j)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "df" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "。"] }), "\n", _jsxs(_components.h4, { id: "答疑-1", children: [_jsx(_components.a, { className: "heading-link", href: "#答疑-1", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "答疑"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "问" }), "：为什么一定要在最外层枚举 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "？"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "答" }), "：仔细看上面的状态转移方程，要正确地算出 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][i][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "，必须先把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "、", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][k][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 算出来。由于我们不知道 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i,j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的大小关系，只有把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 放在最外层枚举，才能保证先把 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mtext, { children: "、" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][j]、f[k][i][k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "、" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][k][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 算出来。顺带一提，对于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 来说，由于在计算 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), 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marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), " 的时候，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, 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_jsx(_components.span, { className: "token keyword", children: "if" }), " cnt ", _jsx(_components.span, { className: "token operator", children: "<=" }), " min_cnt", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token comment", children: "# 相等时取最大的 i" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" min_cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " cnt\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ans ", _jsx(_components.span, { className: "token operator", children: "=" }), " i\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ans\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "复杂度分析-2", children: [_jsx(_components.a, { className: "heading-link", href: "#复杂度分析-2", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "复杂度分析"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["时间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["空间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "五空间优化", children: [_jsx(_components.a, { className: "heading-link", href: "#五空间优化", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "五、空间优化"] }), "\n", _jsxs(_components.p, { children: ["观察上面的状态转移方程，在计算 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), " 时，只会用到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "，不会用到（第一个维度的）下标小于 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 的状态。"] }), "\n", _jsxs(_components.p, { children: ["能不能像", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 背包那样，把第一个维度去掉呢？也就是"] }), "\n", _jsx(_components.p, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[i][j]=min(f[i][j],f[i][k]+f[k][j])" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "min" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "])" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["不幸的是，我们无法确定 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i,j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的大小关系，上式中的", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[i][k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "的值可能是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][i][k]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), "，也有可能被覆盖成了 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][i][k]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "。同样的，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 的值也可能被覆盖成了 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][k][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] })] }), "\n", _jsx(_components.p, { children: "但是！从状态的定义来看：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][i][k]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), " 表示从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), " 的最短路长度，并且这条最短路的中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] })] })] }), "。由于终点是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "，那么中间节点必然不包含 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }), "，所以中间节点编号都 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "≤k−1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "，所以 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][i][k]=f[k][i][k]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "！"] }), "\n", _jsxs(_components.li, { children: ["同理，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[k+1][k][j]=f[k][k][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "因为值没变，所以不用担心被覆盖，直接去掉第一个维度。" }), "\n", _jsxs(_components.p, { children: ["初始值 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "w" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[i][j]=w[i][j]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["三重循环结束后，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f[i][j]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 就是从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "j" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }), " 的最短路了。"] }), "\n", _jsx(_components.pre, { className: "language-py", children: _jsxs(_components.code, { className: "language-py code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "class" }), " ", _jsx(_components.span, { className: "token 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_jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " distanceThreshold", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" w ", _jsx(_components.span, { className: "token operator", children: "=" }), 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className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " wt\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" f ", _jsx(_components.span, { className: "token operator", children: "=" }), " w\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " k ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " i ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " j ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# f[i][j] = min(f[i][j], f[i][k] + f[k][j])" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 手动 if 比大小更快" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" s ", _jsx(_components.span, { className: "token operator", children: "=" }), " f", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "k", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " f", _jsx(_components.span, { className: "token punctuation", children: "[" }), "k", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "j", _jsx(_components.span, { className: "token punctuation", children: "]" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " s ", _jsx(_components.span, { className: "token operator", children: "<" }), " f", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "j", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" f", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token 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builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " j ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " j ", _jsx(_components.span, { className: "token operator", children: "!=" }), " i ", _jsx(_components.span, { className: "token keyword", children: "and" }), " f", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "j", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "<=" }), " distanceThreshold", _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" cnt ", _jsx(_components.span, { className: "token operator", children: "+=" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " cnt ", _jsx(_components.span, { className: "token operator", children: "<=" }), " min_cnt", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token comment", children: "# 相等时取最大的 i" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" min_cnt ", _jsx(_components.span, { className: "token operator", children: "=" }), " cnt\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ans ", _jsx(_components.span, { className: "token operator", children: "=" }), " i\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ans\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "复杂度分析-3", children: [_jsx(_components.a, { className: "heading-link", href: "#复杂度分析-3", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "复杂度分析"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["时间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["空间复杂度：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{2})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "思考题", children: [_jsx(_components.a, { className: "heading-link", href: "#思考题", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "思考题"] }), "\n", _jsxs(_components.p, { children: ["向图中添加一条边，如何维护 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "f" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" })] }) })] }), " 数组？"] }), "\n", _jsxs(_components.p, { children: ["最暴力的做法是，每次添加一条边，就用 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^{3})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 时间全部重算一遍。有没有更快的做法呢？"] }), "\n", _jsxs(_components.p, { children: ["这题是 ", _jsx(_components.a, { href: "https://leetcode.cn/problems/design-graph-with-shortest-path-calculator/", target: "_blank", rel: "noopener noreferrer", children: "2642. 设计可以求最短路径的图类" })] }), "\n", _jsxs(_components.h2, { id: "分类题单", children: [_jsx(_components.a, { className: "heading-link", href: "#分类题单", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "分类题单"] }), "\n", _jsxs(_components.details, { "data-callout": true, "data-callout-type": "abstract", children: ["\n", _jsx(_components.summary, { "data-callout-title": true, children: "题单" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsx(_components.p, { children: _jsx(_components.a, { href: "https://leetcode.cn/circle/discuss/RvFUtj/", target: "_blank", rel: "noopener noreferrer", children: "如何科学刷题？" }) }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://leetcode.cn/circle/discuss/0viNMK/", target: "_blank", rel: "noopener noreferrer", children: 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MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };