"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", code: "code", h3: "h3", img: "img", math: "math", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", msup: "msup", p: "p", path: "path", pre: "pre", semantics: "semantics", span: "span", strong: "strong", svg: "svg", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.p, { children: ["即给定一个集合，给出所有的排列情况\n解法是", _jsx(_components.a, { href: "/%E5%8E%9F%E5%AD%90%E5%B1%82%E7%9F%A5%E8%AF%86%E7%82%B9/%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%F0%9F%90%B8", className: "internal", "data-fs-resolved-file-path": "/原子层知识点/回溯算法🐸.md", children: "回溯算法🐸" })] }), "\n", _jsxs(_components.h3, { id: "无重复元素时", children: [_jsx(_components.a, { className: "heading-link", href: "#无重复元素时", 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" }) }) }), "无重复元素时"] }), "\n", _jsxs(_components.p, { children: ["为了实现每个元素只被选择一次，我们", _jsx(_components.strong, { children: "不用" }), "在每次选择时都检测本次的元素是否在已排列的数字中，我们考虑引入一个布尔型数组 ", _jsx(_components.code, { className: "language-auto", children: "selected" }), " ，其中 ", _jsx(_components.code, { className: "language-auto", children: "selected[i]" }), " 表示 ", _jsx(_components.code, { className: "language-auto", children: "choices[i]" }), " 是否已被选择，并基于它实现以下剪枝操作。"] }), "\n", _jsx(_components.pre, { className: "language-cpp", children: _jsxs(_components.code, { className: "language-cpp code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "/* 回溯算法：全排列 I */" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token operator", children: "&" }), "state", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", 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_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 punctuation", children: ";" }), "\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 comment", children: "// 遍历所有选择" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " i ", 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_jsx(_components.span, { className: "token keyword", children: "int" }), " choice ", _jsx(_components.span, { className: "token operator", children: "=" }), " choices", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 剪枝：不允许重复选择元素" }), "\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 operator", children: "!" }), "selected", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token 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className: "token function", children: "push_back" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "choice", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 进行下一轮选择" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "state", _jsx(_components.span, { className: "token punctuation", children: "," }), " choices", _jsx(_components.span, { className: "token punctuation", children: "," }), " selected", _jsx(_components.span, { className: "token punctuation", children: "," }), " res", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 回退：撤销选择，恢复到之前的状态" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" selected", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token boolean", children: "false" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" state", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "pop_back" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _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 punctuation", children: "}" }), "\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"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "/* 全排列 I */" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["vector", _jsx(_components.span, { className: "token operator", children: "<" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">>" }), " ", _jsx(_components.span, { className: "token function", children: "permutationsI" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " nums", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token 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punctuation", children: "," }), " ", _jsx(_components.span, { className: "token boolean", children: "false" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" vector", _jsx(_components.span, { className: "token operator", children: "<" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">>" }), " res", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "state", _jsx(_components.span, { 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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" }) }) }), "有重复元素时"] }), "\n", _jsxs(_components.p, { children: ["需要进行", _jsx(_components.strong, { children: "相等元素剪枝" }), "，即在本次遍历可能性时，对于重复的元素，只遍历一次\n", _jsx(_components.img, { src: "https://www.hello-algo.com/chapter_backtracking/permutations_problem.assets/permutations_ii_pruning.png", alt: "剪枝", className: "internal" }), "\n在上一题的代码的基础上，我们考虑在每一轮选择中开启一个哈希集合 ", _jsx(_components.code, { className: "language-auto", children: "duplicated" }), " ，用于记录该轮中已经尝试过的元素，并将重复元素剪枝"] }), "\n", _jsx(_components.pre, { className: "language-cpp", children: _jsxs(_components.code, { className: "language-cpp code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "/* 回溯算法：全排列 II */" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token operator", children: "&" }), "state", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token 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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 punctuation", children: ";" }), "\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 comment", children: "// 遍历所有选择" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" unordered_set", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " duplicated", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " i ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " i ", _jsx(_components.span, { className: "token operator", children: "<" }), " choices", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "size" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " i", _jsx(_components.span, { className: "token operator", children: "++" }), _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: "int" }), " choice ", _jsx(_components.span, { className: "token operator", children: "=" }), " choices", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 剪枝：不允许重复选择元素且不允许重复选择相等元素" }), "\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 operator", children: "!" }), "selected", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "&&" }), " duplicated", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "find" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "choice", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "==" }), " duplicated", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "end" }), _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 punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 尝试：做出选择，更新状态" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" duplicated", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "emplace" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "choice", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 记录选择过的元素值" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" selected", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token boolean", children: "true" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" state", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "push_back" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "choice", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 进行下一轮选择" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "state", _jsx(_components.span, { className: "token punctuation", children: "," }), " choices", _jsx(_components.span, { className: "token punctuation", children: "," }), " selected", _jsx(_components.span, { className: "token punctuation", children: "," }), " res", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 回退：撤销选择，恢复到之前的状态" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" selected", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token boolean", children: "false" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" state", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "pop_back" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _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 punctuation", children: "}" }), "\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"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "/* 全排列 II */" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["vector", _jsx(_components.span, { className: "token operator", children: "<" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">>" }), " ", _jsx(_components.span, { className: "token function", children: "permutationsII" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " nums", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">" }), " state", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "bool" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token function", children: "selected" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "nums", _jsx(_components.span, { className: "token punctuation", children: "." }), _jsx(_components.span, { className: "token function", children: "size" }), _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 boolean", children: "false" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" vector", _jsx(_components.span, { className: "token operator", children: "<" }), "vector", _jsx(_components.span, { className: "token operator", children: "<" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: ">>" }), " res", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "backtrack" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "state", _jsx(_components.span, { className: "token punctuation", children: "," }), " nums", _jsx(_components.span, { className: "token punctuation", children: "," }), " selected", _jsx(_components.span, { className: "token punctuation", children: "," }), " res", _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" }), " res", _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: "效率分析", children: [_jsx(_components.a, { className: "heading-link", href: "#效率分析", 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" }) }) }), "效率分析"] }), "\n", _jsxs(_components.p, { children: ["假设元素两两之间互不相同，则 n 个元素共有 n! 种排列（阶乘）；在记录结果时，需要复制长度为 n 的列表，使用", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 时间。", _jsxs(_components.strong, { children: ["因此时间复杂度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "!" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n!n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: "!" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] })] }), " 。"] }), "\n", _jsxs(_components.p, { children: ["最大递归深度为 n ，使用 O(n) 栈帧空间。", _jsx(_components.code, { className: "language-auto", children: "selected" }), " 使用 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 空间。同一时刻最多共有 n 个 ", _jsx(_components.code, { className: "language-auto", children: "duplicated" }), " ，使用 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^2)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 空间。", _jsxs(_components.strong, { children: ["因此空间复杂度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n^2)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] })] })] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };

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