"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", br: "br", code: "code", div: "div", em: "em", h2: "h2", h3: "h3", h4: "h4", img: "img", li: "li", mark: "mark", math: "math", mfrac: "mfrac", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", mstyle: "mstyle", msub: "msub", msubsup: "msubsup", mtable: "mtable", mtd: "mtd", mtext: "mtext", mtr: "mtr", ol: "ol", p: "p", path: "path", pre: "pre", semantics: "semantics", span: "span", strong: "strong", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.h2, { id: "31-栈", children: [_jsx(_components.a, { className: "heading-link", href: "#31-栈", 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" }) }) }), "3.1 栈"] }), "\n", _jsxs(_components.h3, { id: "311-栈的基本概念", children: [_jsx(_components.a, { className: "heading-link", href: "#311-栈的基本概念", 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" }) }) }), "3.1.1 栈的基本概念"] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "栈的定义" }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "栈" }), "（Stack）是", _jsx(_components.strong, { children: "只允许在一端" }), "进行插入或删除操作的线性表。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "栈顶" }), "（Top）：线性表", _jsx(_components.strong, { children: "允许进行插入和删除操作" }), "的那一端。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "栈底" }), "（Bottom）：固定的，", _jsx(_components.strong, { children: "不允许进行插入和删除操作" }), "的另一端。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "空栈" }), "：不含任何元素的空表。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["栈的操作特性可以明显地概括为：", _jsx(_components.strong, { children: "后进先出" }), "（", _jsx(_components.em, { children: "Last in first Out，LIFO" }), "）。"] }), "\n"] }), "\n", _jsxs(_components.ol, { start: "2", children: ["\n", _jsx(_components.li, { children: "栈的基本操作" }), "\n"] }), "\n", _jsx(_components.p, { children: "严蔚敏教材中给出的基本操作：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "InitStack(&S)" }), "：初始化一个空栈。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "StackEmpty(S)" }), "：判断一个栈是否为空，若栈S为空则返回true，否则返回false。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "Push(&S, x)" }), "：入栈，若栈S未满，则降x加入使之成为新栈顶。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "Pop(&S, &x)" }), "：出栈，若栈S非空，则弹出栈顶元素，并用x返回。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "GetTop(S, &x)" }), "：读栈顶元素，但不出栈，若栈S非空，则用x返回栈顶元素。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "DestroyStack(&S)" }), "：销毁栈，并释放栈S占用的存储空间（“&”表示引用调用）。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "在解算法题时若题目未作限制，则也可以直接使用这些基本的操作函数。" }), "\n"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "栈的数学性质" }), "：当 ", _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: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 个不同元素入栈时，出栈元素不同排列的个数为 ", _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: [_jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.msubsup, { children: [_jsx(_components.mi, { children: "C" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{1}{n+1}C_{2n}^{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2484em", verticalAlign: "-0.4033em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8451em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.4033em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: [_jsxs(_components.span, { style: { top: "-2.4519em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }), _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: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2481em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] })] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "该公式称为*卡特兰数(Catalan)*公式，可用数学归纳法证明。" }), "\n"] }), "\n", _jsxs(_components.h3, { id: "312-栈的顺序存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#312-栈的顺序存储结构", 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" }) }) }), "3.1.2 栈的顺序存储结构"] }), "\n", _jsxs(_components.h4, { id: "1-顺序栈的实现", children: [_jsx(_components.a, { className: "heading-link", href: "#1-顺序栈的实现", 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" }) }) }), "1. 顺序栈的实现"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["采用顺序存储结构的栈称为", _jsx(_components.em, { children: "顺序栈" }), "，它利用一组", _jsx(_components.strong, { children: "地址连续的存储单元" }), "存放自栈底到栈顶到数据元素，同时附设一个", _jsx(_components.em, { children: "指针（top）" }), " 指示当前栈顶元素的位置。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "看到顺序栈要想到其顺序存储，也就是地址是连续存放的。和顺序表有一定的相似之处。" }), "\n"] }), "\n", _jsx(_components.p, { children: "栈的顺序存储类型的定义为：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsxs(_components.span, { className: "token macro property", children: [_jsx(_components.span, { className: "token directive-hash", children: "#" }), _jsx(_components.span, { className: "token directive keyword", children: "define" }), " ", _jsx(_components.span, { className: "token macro-name", children: "MaxSize" }), " ", _jsx(_components.span, { className: "token expression", children: _jsx(_components.span, { className: "token number", children: "50" }) })] }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "MaxSize", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 存放栈中元素" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " top", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 栈顶指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["栈顶：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈顶指针" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "初始设置" }), " ", _jsx(_components.code, { className: "language-auto", children: "S.top=-1" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈顶元素" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.data[S.top]" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["出栈入栈：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "入栈操作" }), "：栈不满时，栈顶指针先加1，再送值到栈顶。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "出栈操作" }), "：栈非空时，先取栈顶元素，再将栈顶指针减1."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["判空和判满：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈空条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top==-1" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈满条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top==MaxSize-1" }), "；"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["长度：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈长" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top+1" }), "。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "另一种常见方式：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "初始设置" }), "栈顶指针 ", _jsx(_components.code, { className: "language-auto", children: "S.top=0" }), ";（⚠️注意与前一种方法区分）"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "入栈" }), "时先将值送到栈顶，栈顶指针加1。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "出栈" }), "时，栈顶指针先减1，再取栈顶元素。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈空条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top==0" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "栈满条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "S.top==MaxSize" }), "。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["第一种方法中，", _jsx(_components.code, { className: "language-auto", children: "S.top" }), " 时刻指向栈顶元素，而第二种方法更加有先见之明，时刻指向那个可以插入新元素的末尾空位置。"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "warning", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "栈的判空、判满条件会根据实际给出的条件不同发生变化，下文中代码实现是在栈顶指针初始化为-1的条件下的相应方法，其他情况需具体问题具体分析。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-顺序栈道基本操作", children: [_jsx(_components.a, { className: "heading-link", href: "#2-顺序栈道基本操作", 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" }) }) }), "2. 顺序栈道基本操作"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["这里的 ", _jsx(_components.code, { className: "language-auto", children: "S.top" }), " 时刻指向栈顶元素。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "（1）初始化" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InitStack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqStack ", _jsx(_components.span, { className: "token operator", children: "&" }), "S", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 初始化栈顶指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), " \n"] })] }) }), "\n", _jsx(_components.p, { children: "（2）栈判空" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "StackEmpty" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqStack S", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 如果是-1就是空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), " ", _jsx(_components.span, { className: "token keyword", children: "else" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 不空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（3）入栈" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "Push" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqStack ", _jsx(_components.span, { className: "token operator", children: "&" }), "S", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType x", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "==" }), "MaxSize", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 如果栈满，报错" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "." }), "data", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token operator", children: "++" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "=" }), "x", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 先移动指针，再存放值" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（4）出栈" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "Pop" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqStack ", _jsx(_components.span, { className: "token operator", children: "&" }), "S", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType ", _jsx(_components.span, { className: "token operator", children: "&" }), "x", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 栈空还想出栈？报错！ " }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), " \n"] }), _jsxs(_components.span, { className: "code-line", children: [" x", _jsx(_components.span, { className: "token operator", children: "=" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "--" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 先出栈，指针再减 1" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（5）读栈顶元素" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "GetTop" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqStack S", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType ", _jsx(_components.span, { className: "token operator", children: "&" }), "x", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 栈空读空气？" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x", _jsx(_components.span, { className: "token operator", children: "=" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "S", _jsx(_components.span, { className: "token punctuation", children: "." }), "top", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 读取栈顶元素->x" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "3-共享栈", children: [_jsx(_components.a, { className: "heading-link", href: "#3-共享栈", 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" }) }) }), "3. 共享栈"] }), "\n", _jsx(_components.p, { children: "利用栈底位置不变的特性，可让两个顺序栈共享一个一维数组，将两个栈的栈底分别设置在共享空间的两端，两个栈顶向共享空间中的中间延伸。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["两个栈的栈顶指针都指向栈顶元素；", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "top0=-1" }), " 时0号栈为空；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "top1=MaxSize" }), " 时1号栈为空；"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["仅当两个栈顶指针相邻（", _jsx(_components.code, { className: "language-auto", children: "top1-top0=1" }), "）时，判断为栈满。"] }), "\n", _jsxs(_components.li, { children: ["当0号栈入栈时 ", _jsx(_components.code, { className: "language-auto", children: "top0" }), " 先加1再赋值，1号栈入栈时 ", _jsx(_components.code, { className: "language-auto", children: "top1" }), " 先减1再赋值；出栈时则刚好相反。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "意义" }), "：共享栈是为了更有效利用存储空间，两个栈的空间相互调节，只有在整个存储空间被占满时才发生上溢。其存取数据的时间复杂度仅为O(1)，所以对存取效率没有什么影响。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "313-栈的链式存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#313-栈的链式存储结构", 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" }) }) }), "3.1.3 栈的链式存储结构"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "链栈" }), "：采用链式存储的栈称为", _jsx(_components.em, { children: "链栈" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "优点" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "便于多个栈共享存储空间和提高效率。" }), "\n", _jsx(_components.li, { children: "不存在栈满上溢的情况。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "实现" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "采用单链表实现链栈。" }), "\n", _jsxs(_components.li, { children: ["规定", _jsx(_components.strong, { children: "所有操作" }), "都是在单链表的", _jsx(_components.strong, { children: "表头进行" }), "的。"] }), "\n", _jsxs(_components.li, { children: ["规定链栈", _jsx(_components.strong, { children: "没有头结点" }), "，", _jsx(_components.code, { className: "language-auto", children: "Lhead" }), " 指向", _jsx(_components.strong, { children: "栈顶元素" }), "。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "栈的链式存储结构描述为：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "Linknode" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 数据域" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "Linknode" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "next", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 指针域" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "LiStack", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 栈类型定义" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: ["采用链式存储结构的", _jsx(_components.em, { children: "特点" }), "："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "便于结点的插入和删除。" }), "\n", _jsx(_components.li, { children: "入栈和出栈操作都在链表的表头进行。" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["对于带", _jsx(_components.strong, { children: "头结点" }), "和", _jsx(_components.strong, { children: "不带头结点" }), "的链栈，具体实现方式不同。这点与链表中是一样的，带头结点的链表实现方式也与不带的不同。"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "32-队列", children: [_jsx(_components.a, { className: "heading-link", href: "#32-队列", 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" }) }) }), "3.2 队列"] }), "\n", _jsxs(_components.h3, { id: "321-队列的基本概念", children: [_jsx(_components.a, { className: "heading-link", href: "#321-队列的基本概念", 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" }) }) }), "3.2.1 队列的基本概念"] }), "\n", _jsxs(_components.h4, { id: "1-队列的定义", children: [_jsx(_components.a, { className: "heading-link", href: "#1-队列的定义", 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" }) }) }), "1. 队列的定义"] }), "\n", _jsx(_components.p, { children: "队列（Queue）：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["简称", _jsx(_components.em, { children: "队" }), "。"] }), "\n", _jsx(_components.li, { children: "同为一种操作受限的线性表。" }), "\n", _jsx(_components.li, { children: "只允许在表一端进行插入，在表的另一端进行删除。" }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "插入" }), "：向队列中插入元素称为", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "入队" }), "（或）"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "进队" }), "；"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "删除" }), "：删除元素称为", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "出队" }), "（或）"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "离队" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队头" }), "（", _jsx(_components.em, { children: "Front" }), "）：允许删除的一端，也称", _jsx(_components.em, { children: "队首" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队尾" }), "（", _jsx(_components.em, { children: "Rear" }), "）：允许插入的一端。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "空队列" }), "：不含任何元素的空表。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "直接理解为日常生活中排队就行。最早排队的也是最早离队的。最前面排队的人就是队头的人，只允许你走，你若是插到队头去就算插队了，不是很文明，故也是不被允许的操作。" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["操作特性：", _jsx(_components.mark, { children: "先进先出" }), "（", _jsx(_components.em, { children: "First In First Out，FIFO" }), "）。"] }), "\n", _jsxs(_components.h4, { id: "2-队列常见的基本操作", children: [_jsx(_components.a, { className: "heading-link", href: "#2-队列常见的基本操作", 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" }) }) }), "2. 队列常见的基本操作"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "InitQueue(&Q)" }), "：初始化队列，构造一个空队列Q。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "QueueEmpty(Q)" }), "：判队列空，若队列Q为空返回true，否则返回false。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "EnQueue(&Q, x)" }), "：入队，若队列Q未满，将x加入，使之成为新的队尾。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "DeQueue(&Q, &x)" }), "：出队，若队列Q非空，删除队首元素，并用x返回。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "GetHead(Q, &x)" }), "：读队首元素，若队列Q非空，则将队首元素赋值给x。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["由于", _jsx(_components.strong, { children: "栈" }), "和", _jsx(_components.strong, { children: "队列" }), "是受限制的，因此不是任何队线性表的操作都可以作为栈和队列的操作。比如，不可以随便读取栈或队列", _jsx(_components.em, { children: "中间" }), "的某个元素。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "322-队列的顺序存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#322-队列的顺序存储结构", 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" }) }) }), "3.2.2 队列的顺序存储结构"] }), "\n", _jsxs(_components.h4, { id: "1-队列的顺序存储", children: [_jsx(_components.a, { className: "heading-link", href: "#1-队列的顺序存储", 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" }) }) }), "1. 队列的顺序存储"] }), "\n", _jsx(_components.p, { children: "队列的顺序实现是指分配一块连续的存储单元存放队列的元素，并附设两个指针：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队首指针" }), _jsx(_components.code, { className: "language-auto", children: "front" }), " 指向队首元素"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队尾指针" }), _jsx(_components.code, { className: "language-auto", children: "rear" }), " 指向队尾元素的", _jsx(_components.em, { children: "下一个位置" })] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "不同教材对指针的规定不同，其实现的具体操作也就不同。但归根结底，其思想都是一致的，请结合具体情况具体分析。" }), "\n"] }), "\n", _jsx(_components.p, { children: "队列的顺序存储类型是可以这样描述：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsxs(_components.span, { className: "token macro property", children: [_jsx(_components.span, { className: "token directive-hash", children: "#" }), _jsx(_components.span, { className: "token directive keyword", children: "define" }), " ", _jsx(_components.span, { className: "token macro-name", children: "MaxSize" }), " ", _jsxs(_components.span, { className: "token expression", children: [_jsx(_components.span, { className: "token number", children: "50" }), " "] }), _jsx(_components.span, { className: "token comment", children: "// 定义队列中元素的最大个数" })] }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "MaxSize", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 用数组存放队列元素" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " front", _jsx(_components.span, { className: "token punctuation", children: "," }), "rear", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队首指针和队尾指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), " SqQueue", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "初始时" }), "：", _jsx(_components.code, { className: "language-auto", children: "Q.front=Q.rear=0" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "入队操作" }), "：队不满时，先送值到队尾元素，再将队尾指针加1。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "出队操作" }), "：队不空时，先取队首元素值，再将队首指针加1."] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["这里初始状态的 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), " 可以作为队列的判空条件。"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "warning", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "判断队满需注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: ["能否用 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear==MaxSize" }), " 作为队满的判断条件呢？答案是显然不能，当队列中入队并使得对队列满时，", _jsx(_components.code, { className: "language-auto", children: "rear" }), " 指向了队列能承受最大元素的位置(", _jsx(_components.code, { className: "language-auto", children: "rear=MaxSize" }), ")。此时前面的“人”出队，只留下最后一个人，则 ", _jsx(_components.code, { className: "language-auto", children: "rear" }), " 仍然指向该处，此时 ", _jsx(_components.code, { className: "language-auto", children: "rear=MaxSize" }), " 但显然由于前面的人已经出队，导致此时误判为“溢出”，也就是书上说的假溢出。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-循环队列", children: [_jsx(_components.a, { className: "heading-link", href: "#2-循环队列", 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" }) }) }), "2. 循环队列"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "循环队列" }), "就是为了解决“假溢出”的问题。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "循环队列" }), "：将顺序队列臆造为一个环状的空间（把存储队列元素的表从逻辑上看作一个环）。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "实现" }), "：当队首指针 ", _jsx(_components.code, { className: "language-auto", children: "Q.front=MaxSize-1" }), " 后，再前进一个元素就自动到0（利用除法取模运算实现）。"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "question", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "那么队列的判空条件是什么呢？" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: ["显然是 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), "。但如果入队的速度太快，队尾指针很快就会赶上队首指针，此时尽管队列是满的，但也会出现 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), " 的情况，所以必须采取一些方法来解决："] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "牺牲一个单元来区分队空和队满（比较普遍的做法）。约定“队首指针在队尾下一个位置作为队满的标志”。" }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队满条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "(Q.rear+1)%MaxSize==Q.front" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队空条件" }), "：", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队列中元素的个数" }), "：", _jsx(_components.code, { className: "language-auto", children: "(Q.rear-Q.front+MaxSize)%MaxSize" }), "。"] }), "\n"] }), "\n", _jsxs(_components.ol, { start: "2", children: ["\n", _jsxs(_components.li, { children: ["类型中增设 ", _jsx(_components.code, { className: "language-auto", children: "size" }), " 数据成员，表示元素个数。"] }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若删除成功，则 ", _jsx(_components.code, { className: "language-auto", children: "size" }), " 减1，若插入成功，则 ", _jsx(_components.code, { className: "language-auto", children: "size" }), " 加1。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队空" }), "时，", _jsx(_components.code, { className: "language-auto", children: "Q.size==0" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队满" }), "时，", _jsx(_components.code, { className: "language-auto", children: "Q.size==MaxSize" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "队空和队满都有：" }), " ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), "。"] }), "\n"] }), "\n", _jsxs(_components.ol, { start: "3", children: ["\n", _jsxs(_components.li, { children: ["类型中增设 ", _jsx(_components.code, { className: "language-auto", children: "tag" }), " 数据成员，以区分是队满还是队空。"] }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "删除成功" }), "置 ", _jsx(_components.code, { className: "language-auto", children: "tag=0" }), "，若导致 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), "，则为", _jsx(_components.strong, { children: "队空" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "插入成功" }), "置 ", _jsx(_components.code, { className: "language-auto", children: "tag=1" }), "，若导致 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==Q.rear" }), "，则为", _jsx(_components.strong, { children: "队满" }), "。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["这里队列中的元素个数不是：", _jsx(_components.code, { className: "language-auto", children: "(Q.rear-Q.front)%MaxSize" }), " 的原因是 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear-Q.front" }), " 可能为负数。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "3-循环队列的操作", children: [_jsx(_components.a, { className: "heading-link", href: "#3-循环队列的操作", 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" }) }) }), "3. 循环队列的操作"] }), "\n", _jsx(_components.p, { children: "（1）初始化" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InitQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "=" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _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: "// 初始化队首、队尾指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（2）判队空" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "isEmpty" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqQueue Q", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "==" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队空条件" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "else" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（3）入队" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "EnQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType x", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "+" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "%" }), "MaxSize", _jsx(_components.span, { className: "token operator", children: "==" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队满则报错" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "=" }), "x", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "+" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "%" }), "MaxSize", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队尾指针加1取模" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), " \n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（4）出队" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "DeQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "SqQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType ", _jsx(_components.span, { className: "token operator", children: "&" }), "x", _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" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "==" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队空还想出队呢？" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x", _jsx(_components.span, { className: "token operator", children: "=" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "data", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "+" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "%" }), "MaxSize", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队首指针加1取模" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.h3, { id: "323-队列的链式存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#323-队列的链式存储结构", 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" }) }) }), "3.2.3 队列的链式存储结构"] }), "\n", _jsxs(_components.h4, { id: "1-队列的链式存储", children: [_jsx(_components.a, { className: "heading-link", href: "#1-队列的链式存储", 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" }) }) }), "1. 队列的链式存储"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "链式队列" }), "：队列的链式表示。它实际上是一个同时有队首指针和队尾指针的单链表。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "队首指针" }), "：指向队头结点。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "队尾指针" }), "：指向队尾结点（单链表的最后一个结点）。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "队列的链式存储类型可描述为：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "LinkNode" }), _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 链式队列结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "LinkNode" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "next", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "LinkNode", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 链式队列" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" LinkNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "front", _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token operator", children: "*" }), "rear", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 链式队列的队头和队尾指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "LinkQueue", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: ["不带头结点时，当 ", _jsx(_components.code, { className: "language-auto", children: "Q.front==NULL" }), " 且 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear==NULL" }), " 时，链式队列为空。"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "链式队列为空的判断" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "入队" }), "时：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["建立一个新结点，将新结点插入到链表的尾部，并让 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear" }), " 指向这个新插入的结点（若原队列为空队，则令 ", _jsx(_components.code, { className: "language-auto", children: "Q.front" }), " 也指向该结点）。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "出队" }), "时：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["首先判断队是否为空，若不空，则取出队首元素，将其从链表中删除，并让 ", _jsx(_components.code, { className: "language-auto", children: "Q.front" }), " 指向下一个结点（该结点为最后一个结点，则置 ", _jsx(_components.code, { className: "language-auto", children: "Q.front" }), " 和 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear" }), " 都为NULL）。"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "不难看出，不带头结点的链表队列在操作上往往比较麻烦，因此通常将链式队列设计成一个带头结点的单链表，这样插入和删除草走就统一了。" }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "用单链表表示的链式队列特别适合于数据元素变动比较大的情形，而且不存在队列满且产生溢出的问题。" }), "\n", _jsx(_components.li, { children: "另外，假如程序中要使用多个队列，与多个栈的情形一样，最好使用链式队列，这样就不会出现存储分配不合理和“溢出”的问题。" }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-链式队列的基本操作", children: [_jsx(_components.a, { className: "heading-link", href: "#2-链式队列的基本操作", 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" }) }) }), "2. 链式队列的基本操作"] }), "\n", _jsx(_components.p, { children: "（1）初始化" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InitQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 初始化带头结点的链式队列" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "=" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkNode", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token function", children: "malloc" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "sizeof" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkNode", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), _jsx(_components.span, { className: "token comment", children: "// 建立头结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 初始为空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（2）判队空" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "IsEmpty" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkQueue Q", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "==" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 判空条件" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "else" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（3）入队" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "EnQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType x", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" LinkNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "s", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkNode ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token function", children: "malloc" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "sizeof" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkNode", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), _jsx(_components.span, { className: "token comment", children: "//创建新结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" s", _jsx(_components.span, { className: "token operator", children: "->" }), "data", _jsx(_components.span, { className: "token operator", children: "=" }), "x", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" s", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token operator", children: "=" }), "s", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 插入链尾" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "=" }), "s", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 修改尾指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "（4）出队" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["bool ", _jsx(_components.span, { className: "token function", children: "DeQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "LinkQueue ", _jsx(_components.span, { className: "token operator", children: "&" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "ElemType ", _jsx(_components.span, { className: "token operator", children: "&" }), "x", _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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "==" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " false", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 空队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" LinkNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "p", _jsx(_components.span, { className: "token operator", children: "=" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 用p指向队头结点的后一个结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x", _jsx(_components.span, { className: "token operator", children: "=" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "data", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 取出数据->x" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token operator", children: "=" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "next", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 出队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "==" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 只有一个结点时队头结点的下一个结点p就是队尾结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "rear", _jsx(_components.span, { className: "token operator", children: "=" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "." }), "front", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 若原队只有一个结点，删除后变空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "free" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _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: "return" }), " true", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["这里单独对只有一个结点时做判断是因为只有一个结点时，", _jsx(_components.code, { className: "language-auto", children: "p" }), " 指向空。但向前移动front后 ", _jsx(_components.code, { className: "language-auto", children: "Q.rear" }), " 仍然指向已经被free掉的结点A！这是一个", _jsx(_components.strong, { children: "悬空指针" }), "，会导致严重错误。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "324-双端队列", children: [_jsx(_components.a, { className: "heading-link", href: "#324-双端队列", 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" }) }) }), "3.2.4 双端队列"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "双端队列" }), "：两端都允许进行", _jsx(_components.em, { children: "插入" }), "和", _jsx(_components.em, { children: "删除" }), "操作的线性表。"] }), "\n", _jsxs(_components.li, { children: ["左端：也视为", _jsx(_components.em, { children: "前端" }), "。"] }), "\n", _jsxs(_components.li, { children: ["右端：也视为", _jsx(_components.em, { children: "后端" }), "。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "在双端队列入队时，" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "前端进入的元素" }), "排列在队列中后端进的元素的前面，"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "后端进的元素排列" }), "在队列中前端进的元素的后面。"] }), "\n", _jsxs(_components.li, { children: ["出队时：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "无论是前端还是后端出队，先出的元素排列在后出的元素的前面。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "所以无论你干的是前端还是后端，被裁的时候都是一视同仁。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsxs(_components.div, { "data-callout-title": true, children: [_jsx(_components.strong, { children: "输出受限" }), "的双端队列"] }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "允许在一端进行插入和删除，但另一端只允许插入的双端队列。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsxs(_components.div, { "data-callout-title": true, children: [_jsx(_components.strong, { children: "输入受限" }), "的双端队列"] }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "允许在一端进行插入和删除，但另一端只允许删除的双端队列。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "删除操作被砍了，就是输出操作受限。因为删除就是出队，出队的时候人从队伍中出去那不就是输出了吗。\n同理，插入操作被砍了，就是输入操作受限。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意区分" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "请注意区分双端队列与普通队列的区别，" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "普通队列" }), "两端各负责一个功能"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "双端队列" }), "的一端完全开放，既能插入也能删除，但另一端是受限制的。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "输入受限" }), "的一端完全开放，一端输入受限🚫。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "输出受限" }), "的一端完全开放，一端输出受限🚫。"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "解题技巧" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "栈是退化后的队列，所以栈能完成的操作队列一定能完成。" }), "\n", _jsx(_components.li, { children: "如果要判断一个双端队列是否合法，只需要找到退化成栈后不能完成的那些操作进行检验即可。然后根据序列的相对位置进行拼凑。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "33-栈和队列的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#33-栈和队列的应用", 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" }) }) }), "3.3 栈和队列的应用"] }), "\n", _jsx(_components.p, { children: "想要学会栈和队列，必须学习其应用，把握规律，举一反三。" }), "\n", _jsxs(_components.h3, { id: "331-栈在括号匹配中的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#331-栈在括号匹配中的应用", 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" }) }) }), "3.3.1 栈在括号匹配中的应用"] }), "\n", _jsxs(_components.p, { children: ["假设表达式中允许包含两种括号：圆括号和方括号，其嵌套的顺序任意即 ", _jsx(_components.code, { className: "language-auto", children: "({}())" }), " 或 ", _jsx(_components.code, { className: "language-auto", children: "[([][])]" }), " 等均为正确的格式，", _jsx(_components.code, { className: "language-auto", children: "[(])" }), " 或 ", _jsx(_components.code, { className: "language-auto", children: "([())" }), " 或 ", _jsx(_components.code, { className: "language-auto", children: "(()]" }), " 均为不正确的格式。\n考虑下列括号序列："] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202511212201050.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["分析如下：\n1）计算机接收第一个括号 ", _jsx(_components.code, { className: "language-auto", children: "[" }), " 后，期待与之匹配的第8个括号 ", _jsx(_components.code, { className: "language-auto", children: "]" }), " 出现。\n2）获得了第2个括号 ", _jsx(_components.code, { className: "language-auto", children: "(" }), "，此时第1个括号 ", _jsx(_components.code, { className: "language-auto", children: "[" }), " 暂时放在一边，而急迫期待与之匹配的第7个括号 ", _jsx(_components.code, { className: "language-auto", children: ")" }), " 出现。\n3）后的了第3个括号 ", _jsx(_components.code, { className: "language-auto", children: "[" }), "，此时第2个括号 ", _jsx(_components.code, { className: "language-auto", children: "(" }), " 暂时被放在一边，而迫切期待与之匹配的第4个括号 ", _jsx(_components.code, { className: "language-auto", children: "]" }), " 出现。第3个括号的期待得到满足，消解之后，第2个括号的期待匹配又成为当前最急迫的任务。\n4）以此类推，可见该处理过程与栈道思想吻合。"] }), "\n", _jsx(_components.p, { children: "算法的思路如下：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["初始设置一个空栈，顺序读入括号", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若是", _jsx(_components.strong, { children: "左括号" }), "，则作为一个新的更急迫的期待压入栈中，自然使原有的栈中所有未消解的期待的急迫性降了一级。"] }), "\n", _jsxs(_components.li, { children: ["若是", _jsx(_components.strong, { children: "右括号" }), "，则或使置于栈顶的最急迫期待得以消解，或是不合法的情况（括号序列不匹配，退出程序）。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["算法", _jsx(_components.em, { children: "结束" }), "时，栈为空，否则括号序列不匹配。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "332-栈在表达式求值中的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#332-栈在表达式求值中的应用", 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" }) }) }), "3.3.2 栈在表达式求值中的应用"] }), "\n", _jsx(_components.p, { children: "表达式求值是程序设计语言中一个最基本的问题，它是栈应用的一个典型范例。" }), "\n", _jsxs(_components.h4, { id: "1--算数表达式", children: [_jsx(_components.a, { className: "heading-link", href: "#1--算数表达式", 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" }) }) }), "1 . 算数表达式"] }), "\n", _jsx(_components.p, { children: "中缀表达式（如3+4）是人们常用的算数表达式，操作符中以中缀形式处于操作数的中间。与前缀表达式（如+34）或后缀表达式（如34+）相比，中缀表达式不容易被计算机解析，但仍被许多程序语言使用，因为它更符合人们的思维习惯。" }), "\n", _jsx(_components.p, { children: "与前缀表达式或后缀表达式不同的是，中缀表达式中的括号是必需的。计算过程中必需用括号将操作符和对应的操作数括起来，用于指示运算的次序。后缀表达式的运算符在操作数后面，后缀表达式中考虑了运算符的优先级，没有括号，只有操作数和运算符。" }), "\n", _jsxs(_components.p, { children: ["中缀表达式 ", _jsx(_components.code, { className: "language-auto", children: "A+B*(C-D)-E/F" }), " 对应的后缀表达式为 ", _jsx(_components.code, { className: "language-auto", children: "ABCD-*+EF/-" }), "，将后缀表达式与原表达式对应的表达式树的后序遍历序列进行比较，可发现它们有异曲同工之妙。"] }), "\n", _jsxs(_components.h4, { id: "2--中缀表达式转后缀表达式", children: [_jsx(_components.a, { className: "heading-link", href: "#2--中缀表达式转后缀表达式", 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" }) }) }), "2 . 中缀表达式转后缀表达式"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "中缀表达式转后缀表达式的方法" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["下面先给出一种由中缀表达式转后缀表达式的", _jsx(_components.strong, { children: "手算方法" }), "。\n1）按照运算符的运算顺序", _jsx(_components.mark, { children: "对所有运算单位加括号" }), "\n2）将", _jsx(_components.mark, { children: "运算符移至对应括号的后面" }), "，相当于按“左操作数-右操作符-运算符”重新组合。\n3）", _jsx(_components.mark, { children: "去除所有括号" }), "。"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["例如，中缀表达式 ", _jsx(_components.code, { className: "language-auto", children: "A+B*(C-D)-E/F" }), " 转后缀表达式的过程如下（下标表示运算符的运算顺序）："] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["1）加括号：", _jsx(_components.code, { className: "language-auto", children: "((A+(B*(C-D）))-(E/F))" }), "。\n2）运算符后移：", _jsx(_components.code, { className: "language-auto", children: "((A(B(CD)-)*)+(EF)/)-" }), "。\n3）去除括号后，得到后缀表达式：", _jsx(_components.code, { className: "language-auto", children: "ABCD-*+EF/-" }), "。"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "注意这里是手算的方法，若用计算机实现则只用第一步的表达式。" }), "\n"] }), "\n", _jsx(_components.p, { children: "在计算机中，中缀表达式转后缀表达式时需要借助一个栈，用于保存暂时还不能确定运算顺序的运算符。从左到右依次扫描中缀表达式中的每一项，具体转化过程如下：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["遇到", _jsx(_components.em, { children: "操作数" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "直接加入后缀表达式。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["遇到", _jsx(_components.em, { children: "界限符" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "若为“（”，则直接入栈；" }), "\n", _jsxs(_components.li, { children: ["若为“）”，", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["则", _jsx(_components.em, { children: "不入栈" }), "，"] }), "\n", _jsxs(_components.li, { children: ["且", _jsx(_components.em, { children: "依次弹出栈中的运算符后加入后缀表达式" }), "，"] }), "\n", _jsx(_components.li, { children: "直到遇到“（”为止，" }), "\n", _jsx(_components.li, { children: "并直接删除“（”。" }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["遇到", _jsx(_components.em, { children: "运算符" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["其", _jsx(_components.strong, { children: "优先级高于栈顶运算符" }), "或"] }), "\n", _jsxs(_components.li, { children: ["遇到", _jsx(_components.strong, { children: "栈顶为“（”" }), "，"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.li, { children: "则直接入栈；" }), "\n", _jsxs(_components.li, { children: ["若", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["其", _jsx(_components.strong, { children: "优先级低于或等于栈顶运算符" }), "，"] }), "\n", _jsxs(_components.li, { children: ["则依次弹出栈中的运算符并加入后缀表达式，", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.strong, { children: "直到遇到一个优先级低于它的运算符" }) }), "\n", _jsxs(_components.li, { children: ["或 ", _jsx(_components.strong, { children: "遇到“（”" })] }), "\n", _jsxs(_components.li, { children: ["或", _jsx(_components.strong, { children: "栈空" }), "为止，"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.li, { children: "之后将当前运算符入栈。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "GPT给出的理解方法" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: ["🔥 最关键的理解：\n", _jsx(_components.strong, { children: "栈里存的永远是“还不能决定顺序的运算符”。" }), _jsx(_components.br, {}), "\n", _jsx(_components.strong, { children: "后缀表达式中出现的永远是“已经确定执行顺序的符号”。" })] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "只要你抓住这两句话，你完全不需要死记步骤，会“自然地知道操作”。" }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "记忆心法🌈" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "language-auto code-highlight", children: [_jsx(_components.span, { className: "code-line", children: "操作数 → 出\n" }), _jsx(_components.span, { className: "code-line", children: "( → 入\n" }), _jsx(_components.span, { className: "code-line", children: ") → 弹到 (\n" }), _jsx(_components.span, { className: "code-line", children: "op → 弹掉所有 >= 它的，再入栈> \n" })] }) }), "\n"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "栈的深度分析" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "栈的深度" }), "：指栈中元素的个数。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "通常" }), "是给出入栈和出栈道序列，求最大深度（栈的容量当然应大于或等于最大深度）。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "有时" }), "会间接给出入栈和出栈序列，例如以中缀表达式和后缀表达式的形式给出入栈和出栈序列。"] }), "\n", _jsxs(_components.li, { children: ["掌握栈道", _jsx(_components.em, { children: "先进后出的特点" }), "进行手工模拟是解决这类问题的有效方法。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "跳出题目的框架，抓住本质规律才能快速解题。" }), "\n"] }), "\n", _jsxs(_components.h4, { id: "3--后缀表达式求值", children: [_jsx(_components.a, { className: "heading-link", href: "#3--后缀表达式求值", 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" }) }) }), "3 . 后缀表达式求值"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "用栈实现表达式求值的分析" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["通过后缀表示计算表达式值的过程：", _jsx(_components.em, { children: "从左往右" }), "依次扫描表达式中的每一项。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若该项是", _jsx(_components.strong, { children: "操作数" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "则将其压入栈中；" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["若该项是", _jsx(_components.strong, { children: "操作符" }), " ", _jsx(_components.code, { className: "language-auto", children: "" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["则从栈中退出两个操作数 ", _jsx(_components.code, { className: "language-auto", children: "X" }), " 和 ", _jsx(_components.code, { className: "language-auto", children: "Y" }), "，形成运算指令 ", _jsx(_components.code, { className: "language-auto", children: "XY" }), "，"] }), "\n", _jsx(_components.li, { children: "并将结果压入栈中。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["当所有项都", _jsx(_components.strong, { children: "扫描并处理完后" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "栈顶存放的就是最后的计算结果。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "333-栈在递归中的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#333-栈在递归中的应用", 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" }) }) }), "3.3.3 栈在递归中的应用"] }), "\n", _jsxs(_components.p, { children: ["递归是一种重要的程序设计方法。简单地说，若在一个函数、过程或数据结构的定义中又应用了它自身，则这个函数、过程或数据结构称为是", _jsx(_components.em, { children: "递归定义" }), "的，简称", _jsx(_components.em, { children: "递归" }), "。"] }), "\n", _jsxs(_components.p, { children: ["递归通常把一个", _jsx(_components.em, { children: "大型的复杂问题" }), "层层转化为一个", _jsx(_components.em, { children: "与原问题相似的规模较小的问题" }), "来求解，递归策略只需少量的代码就可以描述出解题过程所需要的多次重复计算，大大减少了程序的代码量。但在通常情况下，它的效率并不是太高。"] }), "\n", _jsx(_components.p, { children: "菲波纳契数列就是一个经典的例子，其定义如下：" }), "\n", _jsx(_components.span, { className: "katex-display", 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", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "F" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "{" }), _jsxs(_components.mtable, { rowspacing: 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"1.91em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "arraycolsep", style: { width: "1em" } }), _jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "2.41em" }, children: [_jsxs(_components.span, { style: { top: "-4.41em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] }), 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_jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.91em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "用程序实现：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "int" }), " ", _jsx(_components.span, { className: "token function", children: "F" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " 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" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _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 punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 边界条件" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), " ", _jsx(_components.span, { className: "token keyword", children: "else" }), " ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _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: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 边界条件" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "else" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token function", children: "F" }), _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: ")" }), _jsx(_components.span, { className: "token operator", children: "+" }), _jsx(_components.span, { className: "token function", children: "F" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "n", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "2" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归表达式" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "必需注意递归模型不能是循环定义的，必须满足以下两个条件：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "递归表达式（递归体）" }), "\n", _jsx(_components.li, { children: "边界条件（递归出口）" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "递归的精髓在于降低问题规模。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "栈在函数调用中的作用和工作原理" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "在递归调用的过程中，系统为每一层的返回点、局部变量、传入实惨等开辟了递归工作栈来进行数据存储，递归次数过多容易造成栈溢出等。" }), "\n", _jsxs(_components.li, { children: ["而其", _jsx(_components.strong, { children: "效率不高" }), "的原因是递归调用过程中包含很多重复的计算。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "图见书上P93的图3.16" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["显然，在递归调用的过程中，", _jsx(_components.code, { className: "language-auto", children: "F(3)" }), " 被计算2次，", _jsx(_components.code, { className: "language-auto", children: "F(2)" }), " 被计算3次。", _jsx(_components.code, { className: "language-auto", children: "F(1)" }), " 被调用5次，", _jsx(_components.code, { className: "language-auto", children: "F(0)" }), " 被调用3次。所以，递归的效率低下，但优点是代码简单，容易理解。在第5章第数中利用了递归的思想额，代码变得非常简单。通常情况下，初学者很难理解递归的调用过程，如果想理解递归具体是如何实现的，可以参阅编译原理教材中的相关内容。"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["通过", _jsx(_components.strong, { children: "栈" }), "可以将递归算法转换为非递归算法。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "334-队列在层次遍历中的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#334-队列在层次遍历中的应用", 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" }) }) }), "3.3.4 队列在层次遍历中的应用"] }), "\n", _jsx(_components.p, { children: "在信息处理中有一大类问题需要逐层或逐行处理。这类问题的解决方法往往是在处理当前层或当前行时就对下一层或下一行做预处理，把处理顺序安排好，等到当前层或当前行处理完毕，就可以处理下一层或下一行。使用队列是为了保存下一步的处理顺序。下面用二叉树层次遍历的例子，说明队列的应用。" }), "\n", _jsx(_components.p, { children: "该过程的描述：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "若队空（所有结点都已处理完毕），则结束遍历；否则重复3操作。" }), "\n", _jsxs(_components.li, { children: ["队列中第一个结点出队，并访问之。", "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "若其有左孩子，则将左孩子入队；" }), "\n", _jsx(_components.li, { children: "若其有右孩子，则将右孩子入队，返回第2步。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "335-队列在计算机系统中的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#335-队列在计算机系统中的应用", 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" }) }) }), "3.3.5 队列在计算机系统中的应用"] }), "\n", _jsx(_components.p, { children: "队列在计算机系统中的应用非常广泛，以下仅从两个方面来阐述：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["第一个方面", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["是解决", _jsx(_components.em, { children: "主机与外部设备之间速度不匹配" }), "的问题，"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["第二个方面", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["是解决由", _jsx(_components.em, { children: "多用户引起的资源竞争" }), "问题。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "缓冲区的逻辑结构" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "对于第一个方面，仅以主机和打印机🖨️之间速度不匹配的问题为例做简要说明。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["问题：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "主机输出数据给打印机打印，输出数据的速度比打印机打印数据的速度快得多，因为速度不匹配，若直接把输出的数据送给打印机打印，显然不可行。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["解决方法：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "设置一个打印数据缓冲区，主机把要打印输出的数据依次写入这个缓冲区，写满后就暂停输出，转去做其他事情。打印机就从缓冲区中按照先进先出的原则依次取出数据并打印，打印完后再向主机发出请求。主机接到请求后再向缓冲区写入打印数据。这样做既保证了打印数据的正确，又使主机提高了效率。" }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "由此可见，打印数据缓冲区中所存储的数据就是一个队列。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "多队列出队/入队操作的应用" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "对于第二个方面，CPU资源的竞争是一个典型的例子。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "在一个带有多终端的计算机系统上，有多个用户需要CPU各自运行自己的程序，它们分别通过各自的终端向操作系统提出占用CPU的请求。" }), "\n", _jsxs(_components.li, { children: ["操作系统通常按照每个请求在", _jsx(_components.em, { children: "时间上的先后顺序" }), "，把它们排成一个队列，每次把CPU分配", _jsx(_components.em, { children: "给队首请求的用户使用" }), "。"] }), "\n", _jsxs(_components.li, { children: ["当相应的程序", _jsx(_components.em, { children: "运行结束或用完规定的时间间隔后，令其出队" }), "，再", _jsx(_components.em, { children: "把CPU分配给新的队首请求的用户使用" }), "。"] }), "\n", _jsx(_components.li, { children: "这样既能满足每个用户的请求，又使CPU能够正常运行。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "34-数组和特殊矩阵", children: [_jsx(_components.a, { className: "heading-link", href: "#34-数组和特殊矩阵", 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" }) }) }), "3.4 数组和特殊矩阵"] }), "\n", _jsx(_components.p, { children: "矩阵在计算机图形学、工程计算中占有举足轻重的地位。在数据结构中考虑的是如何用最小的内存空间来存储同样的一组数据。所以，我们不研究矩阵及其运算等，而把精力放在如何将矩阵更有效地存储在内存中，并能方便地提取矩阵中的元素。" }), "\n", _jsxs(_components.h3, { id: "341-数组的定义", children: [_jsx(_components.a, { className: "heading-link", href: "#341-数组的定义", 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" }) }) }), "3.4.1 数组的定义"] }), "\n", _jsxs(_components.p, { children: ["数组是由 ", _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, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n(n\\geq 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", children: "n" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _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", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个", _jsx(_components.strong, { children: "相同类型" }), "的数据元素构成的", _jsx(_components.strong, { children: "有限序列" }), "，每个数据元素称为一个", _jsx(_components.em, { children: "数组元素" }), "，每个元素在 ", _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: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 个线性关系中的", _jsx(_components.em, { children: "序号" }), "称为该", _jsx(_components.strong, { children: "元素的下标" }), "，下标的取值范围称为数组的", _jsx(_components.strong, { children: "维界" }), "。"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.em, { children: "数组" }), "与", _jsx(_components.em, { children: "线性表" }), "的关系："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "数组是线性表的推广。" }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "一维数组" }), "可视为一个", _jsx(_components.em, { children: "线性表" }), "；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "二维数组" }), "可视为其", _jsx(_components.em, { children: "元素是定长数组" }), "的", _jsx(_components.em, { children: "线性表" }), "，以此类推。"] }), "\n", _jsxs(_components.li, { children: ["数组一旦被定义，", _jsx(_components.em, { children: "其维数和维界就不再改变" }), "。"] }), "\n", _jsxs(_components.li, { children: ["因此，除结构的初始化和销毁外，数组只会有", _jsx(_components.em, { children: "存取元素" }), "和", _jsx(_components.em, { children: "修改元素" }), "的操作。"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "342-数组的存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#342-数组的存储结构", 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" }) }) }), "3.4.2 数组的存储结构"] }), "\n", _jsx(_components.p, { children: "大多数计算机语言都提供了数组数据类型，逻辑意义上的数组可采用计算机语言中的数组数据类型进行存储，一个数组的所有元素在内存中占用一段连续的存储空间。" }), "\n", _jsxs(_components.p, { children: ["以一维数组 ", _jsx(_components.code, { className: "language-auto", children: "A[0...n-1]" }), " 为例，其存储结构关系式为："] }), "\n", _jsx(_components.span, { className: "katex-display", 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", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "L" }), _jsx(_components.mi, { children: "O" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mi, { children: "O" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "LOC(a_{i})=LOC(a_{0})+i\\times L,(0\\leq i\\leq n)" })] }) }) }), _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", children: "L" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t 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_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: "L" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", 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: "0" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _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: "0.7429em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _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", children: "L" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _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: "0.7955em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _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: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["其中，", _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: "L" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " 是每个数组元素所占的", _jsx(_components.em, { children: "存储单元" }), "。"] }), "\n", _jsx(_components.p, { children: "对于多维数组，有两种映射方法：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "按行优先" }), "\n", _jsx(_components.li, { children: "按列优先" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["以二维数组为例，", _jsx(_components.strong, { children: "按行优先存储" }), "的基本思想是：先行后列，先存储行号较小的元素，行号相等先存储列号较小的元素。"] }), "\n", _jsxs(_components.p, { children: ["设二维数组的行下标与列下标的范围分别为 ", _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.mo, { stretchy: "false", children: "[" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "[0,h_{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: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "h" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: 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_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "[0,h_{2}]" })] }) }) }), _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: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord 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"mclose", children: "]" })] }) })] }), "，则存储结构关系式为："] }), "\n", _jsx(_components.span, { className: "katex-display", 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", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "L" }), _jsx(_components.mi, { children: "O" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { 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: "L" }), _jsx(_components.mi, { children: "O" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "0" })] })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "LOC(a_{i,j})=LOC(a_{0,0})+[i\\times(h_{2}+1)+j]\\times L" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0361em", verticalAlign: "-0.2861em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { 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_jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _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: "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: "0.0572em" }, children: "j" }), _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: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] })] })] }) }), "\n", _jsx(_components.p, { children: 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children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _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: "mspace", style: { marginRight: 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[_jsx(_components.a, { className: "heading-link", href: "#343-特殊矩阵的压缩矩阵", 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" }) }) }), "3.4.3 特殊矩阵的压缩矩阵"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "压缩矩阵" }), "：指", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["为多个", _jsx(_components.em, { children: "值相同的矩阵只分配一个存储空间" }), "，"] }), "\n", _jsxs(_components.li, { children: ["对", _jsx(_components.em, { children: "零元素不分配空间" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "特殊矩阵" }), "：指", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "具有许多相同矩阵元素或零元素，并且这些相同矩阵元素或零元素的分布有一定规律性的矩阵。" }), "\n", _jsxs(_components.li, { children: ["常见的特殊矩阵有：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "对称矩阵" }), "\n", _jsx(_components.li, { children: "上（下）三角矩阵" }), "\n", _jsx(_components.li, { children: "对角矩阵" }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "特殊矩阵" }), "的", _jsx(_components.strong, { children: "压缩存储方法" }), "：找出特殊矩阵中值相同的矩阵元素的分布规律，把那些呈现规律性分布的、值相同的多个矩阵元素压缩存储到一个存储空间中。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "1-对称矩阵", children: [_jsx(_components.a, { className: "heading-link", href: "#1-对称矩阵", 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" }) }) }), "1. 对称矩阵"] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "对称矩阵" }), "："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若对一个 ", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 阶矩阵 ", _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: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " 中的任意一个元素 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _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: "a_{i,j}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: 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"strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal 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[_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _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: "0.854em", 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" }), _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: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "，则称其为对称矩阵。"] }), "\n", _jsxs(_components.li, { children: ["其中的元素可以划分为3个部分，即", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "上三角区" }), "\n", _jsx(_components.li, { children: "主对角线" }), "\n", _jsx(_components.li, { children: "下三角区。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["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: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 阶对称矩阵，上三角区的所有元素和下三角区的对应元素相同，若仍采用二维数组存放，则会浪费几乎一半的空间，为此将 ", _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: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 阶对称矩阵 ", _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: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " 存放在一维数组 ", _jsx(_components.code, { className: "language-auto", children: "B[n(n+1)/2]" }), " 中，即元素 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _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: "a_{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: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 存放在 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b_{k}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", 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", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 中。"] }), "\n", _jsxs(_components.p, { children: ["比如只存放", _jsx(_components.strong, { children: "下三角" }), "部分（含主对角部线）的元素：\n在数组 ", _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: "B" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "B" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" })] }) })] }), " 中，位于元素 ", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { 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.mi, { children: "i" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{i,j}(i\\geq j)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0361em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _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: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 前面的元素个数为："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["第 ", _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" })] }) })] }), " 行：1个元素（", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{1,1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "）"] }), "\n", _jsxs(_components.li, { children: ["第 ", _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" })] }) })] }), " 行：2个元素（", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "2" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{2,1},a_{2,2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mtight", children: "2" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "）"] }), "\n", _jsx(_components.li, { children: "…" }), "\n", _jsxs(_components.li, { children: ["第 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: 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}) })] }), " 行：", _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: "j" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _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" })] })] })] }), " 个元素（", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: 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_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "）"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["因此，元素 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: 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{ height: "3.01em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "下三角区元素" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.192em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "主对角线上方的元素的关系和前面对称矩阵有所不同。这是因为对称矩阵是填满了的，所以可以直接按照列来计算，而上三角矩阵则为了节省空间按照行优先的原则存储。这就导致需要一行一行地数，所以是一个递减的数列。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "如何理解" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "这部分内容很简单，建议画一个大一点的矩阵自己推导一番，这样就能真正理解这个公式了。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "3-三对角矩阵", children: [_jsx(_components.a, { className: "heading-link", href: "#3-三对角矩阵", 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" }) }) }), "3. 三对角矩阵"] }), "\n", _jsxs(_components.p, { children: ["对角矩阵也叫", _jsx(_components.strong, { children: "带状矩阵" }), "。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "定义" }), "：对 ", _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: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _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: "n" })] }) })] }), " 阶矩阵 ", _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: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " 中任意一个元素 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _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: "a_{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: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "，当 ", _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, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|i-j|>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", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _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: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mord", 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: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 时，若有 ", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{i,j}=0(1\\leq i,j\\leq n)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _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", children: "0" }), _jsx(_components.span, { className: "mopen", children: "(" }), _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: "0.854em", 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" }), _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: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "，则称为三对角矩阵。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "特征" }), "：在三对角矩阵中，所有非零元素都集中在以主对角线为中心的3条对角线的区域，其他区域的元素都为零。"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["三对角矩阵 ", _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: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " 也可以采用压缩存储，将3条对角线上的元素按行优先方式存放在一维数组B中，且 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{1,1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 存放于 ", _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: "B" }), _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: "B[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: "0.0502em" }, children: "B" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }), " 中。"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "也就是一行一行地读取并放入一维数组中。" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["由此可以计算矩阵 ", _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: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " 中3条对角线上的元素 ", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { 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.mn, { children: "1" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { separator: 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className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _jsx(_components.span, { className: "mord", 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", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 在一维数组B中存放的下标为 ", _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: 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_jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7429em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _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: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _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: "3" })] })] })] }), "。"] }), "\n", _jsxs(_components.p, { children: ["反之，若已知三对角矩阵中的某个元素 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "j" })] })] }) }), 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"base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] })] })] }), "，", _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: "j" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j=2-2\\times2+3=1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0572em" }, children: "j" }), _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: "2" }), _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: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _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: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _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: "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: "1" })] })] })] }), "，存放的是 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { 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className: "mord", children: "1" }), _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: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" })] })] })] }), "，", _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: "j" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j=4-2\\times2+3=3" })] }) }) }), _jsxs(_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" }), _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: "4" }), _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: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _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: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _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: "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: "3" })] })] })] }), "，存放的是 ", _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: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "3" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{2,3}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mtight", children: "3" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "。"] }), "\n", _jsx(_components.p, { children: "这一块虽然王道讲的很简略，但实际做题目真的很简单，所以如果觉得难理解（下标的计算细节）可以直接去做一下题试试。" }), "\n", _jsxs(_components.h3, { id: "344-稀疏矩阵", children: [_jsx(_components.a, { className: "heading-link", href: "#344-稀疏矩阵", 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" }) }) }), "3.4.4 稀疏矩阵"] }), "\n", _jsxs(_components.p, { children: ["矩阵中非零元素的个数t，相对矩阵元素的个数s来说非常少，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: "s" }), _jsx(_components.mo, { children: "≫" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "s\\gg t" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _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: "0.6151em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "t" })] })] })] }), " 的矩阵称为", _jsx(_components.strong, { children: "系数矩阵" }), "。"] }), "\n", _jsxs(_components.p, { children: ["例如，一个矩阵的阶为 ", _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.mn, { children: "100" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "100" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "100\\times100" })] }) }) }), _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: "100" }), _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: "100" })] })] })] }), "，该矩阵中的元素少于100个非零元素。"] }), "\n", _jsx(_components.p, { children: "若采用常规的方法存储稀疏矩阵，则相当浪费存储空间，因此仅存储非零元素。但通常非零元素的分布没有规律，所以仅存储非零元素的值是不够的，还要存储它所在的行和列。" }), "\n", _jsxs(_components.p, { children: ["因此，将非零元素及其相应的行和列构成一个三元组（行标i，列标j，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: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _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: "a_{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: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mpunct mtight", children: "," }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0572em" }, children: "j" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "），然后按照某种规律来存储这些三元组线性表。"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "缺点" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["系数矩阵压缩存储后便失去了", _jsx(_components.strong, { children: "随机存取特性" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "存储结构" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "十字链表存储" }), "\n", _jsx(_components.li, { children: "数组存储" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "保存的信息" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "行数" }), "\n", _jsxs(_components.li, { children: ["列数", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "用于判断稀疏矩阵的大小，方便还原。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["非零元素的个数。", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "用于判断稀疏矩阵的稀疏程度。" }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };