{"id":535,"date":"2019-10-15T16:26:20","date_gmt":"2019-10-15T08:26:20","guid":{"rendered":"http:\/\/120.55.184.7\/?p=535"},"modified":"2025-04-15T18:14:40","modified_gmt":"2025-04-15T10:14:40","slug":"535","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=535","title":{"rendered":"\u62d3\u6251\u6392\u5e8f"},"content":{"rendered":"\n

\u62d3\u6251\u6392\u5e8f(Topological Sorting)\u662f\u9488\u5bf9\u6709\u5411\u65e0\u73af\u56fe(DAG, Directed Acyclic Graph)<\/strong>\u7684\u4e00\u79cd\u7ebf\u6027\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u4f7f\u5f97\u5bf9\u4e8e\u56fe\u4e2d\u7684\u6bcf\u4e00\u6761\u6709\u5411\u8fb9(u\u2192v)\uff0c\u9876\u70b9u\u5728\u6392\u5e8f\u4e2d\u90fd\u51fa\u73b0\u5728\u9876\u70b9v\u4e4b\u524d\u3002<\/p>\n\n\n\n

\u4e00\u3001\u6838\u5fc3\u6982\u5ff5\u4e0e\u6027\u8d28<\/h2>\n\n\n\n
    \n
  1. \u57fa\u672c\u5b9a\u4e49<\/strong>\uff1a
    \u2022 \u5bf9\u6709\u5411\u65e0\u73af\u56feG\u8fdb\u884c\u62d3\u6251\u6392\u5e8f\uff0c\u662f\u5c06G\u4e2d\u6240\u6709\u9876\u70b9\u6392\u6210\u4e00\u4e2a\u7ebf\u6027\u5e8f\u5217
    \u2022 \u82e5\u5b58\u5728\u8fb9u\u2192v\uff0c\u5219u\u5728\u5e8f\u5217\u4e2d\u5fc5\u987b\u51fa\u73b0\u5728v\u4e4b\u524d<\/li>\n\n\n\n
  2. \u5173\u952e\u6027\u8d28<\/strong>\uff1a
    \u2022 \u65e0\u73af\u6027<\/strong>\uff1a\u4ec5\u9002\u7528\u4e8eDAG\uff0c\u82e5\u56fe\u4e2d\u5b58\u5728\u73af\u5219\u65e0\u6cd5\u8fdb\u884c\u62d3\u6251\u6392\u5e8f
    \u2022 \u4e0d\u552f\u4e00\u6027<\/strong>\uff1a\u540c\u4e00DAG\u53ef\u80fd\u6709\u591a\u79cd\u5408\u6cd5\u7684\u62d3\u6251\u6392\u5e8f\u7ed3\u679c
    \u2022 \u504f\u5e8f\u5230\u5168\u5e8f<\/strong>\uff1a\u5c06\u56fe\u4e2d\u7684\u504f\u5e8f\u5173\u7cfb\u8f6c\u5316\u4e3a\u5168\u5e8f\u5173\u7cfb<\/li>\n\n\n\n
  3. \u5e94\u7528\u573a\u666f<\/strong>\uff1a
    \u2022 \u4efb\u52a1\u8c03\u5ea6\uff08\u9879\u76ee\u7ba1\u7406\u4e2d\u7684\u4efb\u52a1\u4f9d\u8d56\u5173\u7cfb\uff09
    \u2022 \u8bfe\u7a0b\u5b89\u6392\uff08\u5148\u4fee\u8bfe\u7a0b\u8981\u6c42\uff09
    \u2022 \u7f16\u8bd1\u987a\u5e8f\uff08\u6e90\u6587\u4ef6\u4f9d\u8d56\u5173\u7cfb\uff09
    \u2022 \u7535\u8def\u8bbe\u8ba1\uff08\u5143\u4ef6\u5e03\u7ebf\u987a\u5e8f\uff09
    \u2022 \u751f\u4ea7\u8c03\u5ea6\uff08\u96f6\u90e8\u4ef6\u751f\u4ea7\u987a\u5e8f\uff09<\/li>\n<\/ol>\n\n\n\n

    \u4e8c\u3001\u7b97\u6cd5\u5b9e\u73b0<\/h2>\n\n\n\n

    1. Kahn\u7b97\u6cd5\uff08\u57fa\u4e8e\u5165\u5ea6\uff09<\/h3>\n\n\n\n

    \u7b97\u6cd5\u6b65\u9aa4<\/strong>\uff1a<\/p>\n\n\n\n

      \n
    1. \u8ba1\u7b97\u6240\u6709\u8282\u70b9\u7684\u5165\u5ea6\uff08\u6307\u5411\u8be5\u8282\u70b9\u7684\u8fb9\u6570\uff09<\/li>\n\n\n\n
    2. \u5c06\u5165\u5ea6\u4e3a0\u7684\u8282\u70b9\u52a0\u5165\u961f\u5217<\/li>\n\n\n\n
    3. \u4ece\u961f\u5217\u4e2d\u53d6\u51fa\u8282\u70b9\u5e76\u52a0\u5165\u7ed3\u679c\u5e8f\u5217<\/li>\n\n\n\n
    4. \u5c06\u8be5\u8282\u70b9\u7684\u6240\u6709\u90bb\u63a5\u8282\u70b9\u5165\u5ea6\u51cf1<\/li>\n\n\n\n
    5. \u82e5\u90bb\u63a5\u8282\u70b9\u5165\u5ea6\u53d8\u4e3a0\uff0c\u5219\u52a0\u5165\u961f\u5217<\/li>\n\n\n\n
    6. \u91cd\u590d\u76f4\u5230\u961f\u5217\u4e3a\u7a7a<\/li>\n\n\n\n
    7. \u82e5\u7ed3\u679c\u8282\u70b9\u6570\u2260\u603b\u8282\u70b9\u6570\uff0c\u8bf4\u660e\u6709\u73af<\/li>\n<\/ol>\n\n\n\n

      C++\u5b9e\u73b0<\/strong>\uff1a<\/p>\n\n\n

      \nbool topologicalSort(vector<vector<int>>& G, vector<int>& inDegree, int n) {\n    queue<int> q;\n    vector<int> res;\n\n    for(int i=0; i<n; i++)\n        if(inDegree[i] == 0) q.push(i);\n\n    while(!q.empty()) {\n        int u = q.front(); q.pop();\n        res.push_back(u);\n        for(int v : G[u]) {\n            if(--inDegree[v] == 0)\n                q.push(v);\n        }\n    }\n    return res.size() == n;\n}\n<\/pre><\/div>\n\n\n
      2. DFS\u7b97\u6cd5<\/h3>\n\n\n\n
      \u7b97\u6cd5\u6b65\u9aa4<\/strong>\uff1a<\/p>\n\n\n\n
        \n
      1. \u5bf9\u672a\u8bbf\u95ee\u8282\u70b9\u8fdb\u884cDFS<\/li>\n\n\n\n
      2. \u9012\u5f52\u8bbf\u95ee\u6240\u6709\u90bb\u63a5\u8282\u70b9<\/li>\n\n\n\n
      3. \u9012\u5f52\u5b8c\u6210\u540e\u5c06\u5f53\u524d\u8282\u70b9\u538b\u5165\u6808<\/li>\n\n\n\n
      4. \u6700\u7ec8\u6808\u7684\u9006\u5e8f\u5373\u4e3a\u62d3\u6251\u6392\u5e8f<\/li>\n<\/ol>\n\n\n\n
        Python\u5b9e\u73b0<\/strong>\uff1a<\/p>\n\n\n
        \ndef topological_sort_dfs(graph):\n    visited = set()\n    stack = []\n\n    def dfs(u):\n        visited.add(u)\n        for v in graph.get(u, []):\n            if v not in visited:\n                dfs(v)\n        stack.append(u)\n\n    for u in graph:\n        if u not in visited:\n            dfs(u)\n\n    return stack[::-1]\n<\/pre><\/div>\n\n\n
        \u4e09\u3001\u590d\u6742\u5ea6\u5206\u6790<\/h2>\n\n\n\n
        \u7b97\u6cd5<\/th>\u65f6\u95f4\u590d\u6742\u5ea6<\/th>\u7a7a\u95f4\u590d\u6742\u5ea6<\/th>\u7279\u70b9<\/th><\/tr><\/thead>
        Kahn<\/td>O(V+E)<\/td>O(V)<\/td>\u6613\u4e8e\u7406\u89e3\uff0c\u9002\u5408\u52a8\u6001\u56fe<\/td><\/tr>
        DFS<\/td>O(V+E)<\/td>O(V)<\/td>\u4ee3\u7801\u7b80\u6d01\uff0c\u9700\u8981\u9012\u5f52<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
        \u56db\u3001\u5b9e\u9645\u5e94\u7528\u793a\u4f8b<\/h2>\n\n\n\n
        \u8bfe\u7a0b\u5b89\u6392\u95ee\u9898\uff08LeetCode 207\uff09<\/h3>\n\n\n\n
        \u95ee\u9898<\/strong>\uff1a\u5224\u65ad\u80fd\u5426\u5b8c\u6210\u6240\u6709\u8bfe\u7a0b\uff08\u65e0\u73af\u5219\u53ef\u5b8c\u6210\uff09<\/p>\n\n\n\n
        Java\u89e3\u6cd5<\/strong>\uff1a<\/p>\n\n\n
        \npublic boolean canFinish(int numCourses, int[][] prerequisites) {\n    List<Integer>[] graph = new List[numCourses];\n    int[] inDegree = new int[numCourses];\n\n    for(int i=0; i<numCourses; i++)\n        graph[i] = new ArrayList<>();\n\n    for(int[] cp : prerequisites) {\n        graph[cp[1]].add(cp[0]);\n        inDegree[cp[0]]++;\n    }\n\n    Queue<Integer> q = new LinkedList<>();\n    for(int i=0; i<numCourses; i++)\n        if(inDegree[i] == 0) q.offer(i);\n\n    int count = 0;\n    while(!q.isEmpty()) {\n        int pre = q.poll();\n        count++;\n        for(int cur : graph[pre])\n            if(--inDegree[cur] == 0)\n                q.offer(cur);\n    }\n    return count == numCourses;\n}\n<\/pre><\/div>\n\n\n
        \u4e94\u3001\u5e38\u89c1\u95ee\u9898\u4e0e\u6280\u5de7<\/h2>\n\n\n\n
          \n
        1. \u73af\u68c0\u6d4b<\/strong>\uff1a\u82e5\u62d3\u6251\u6392\u5e8f\u7ed3\u679c\u8282\u70b9\u6570\u5c11\u4e8e\u603b\u8282\u70b9\u6570\uff0c\u8bf4\u660e\u56fe\u4e2d\u5b58\u5728\u73af<\/li>\n\n\n\n
        2. \u5e76\u884c\u4efb\u52a1<\/strong>\uff1a\u540c\u4e00\u5c42\u7ea7\u5165\u5ea6\u4e3a0\u7684\u8282\u70b9\u53ef\u5e76\u884c\u5904\u7406<\/li>\n\n\n\n
        3. \u5b57\u5178\u5e8f\u6392\u5e8f<\/strong>\uff1a\u4f7f\u7528\u4f18\u5148\u961f\u5217\u4ee3\u66ff\u666e\u901a\u961f\u5217\u53ef\u83b7\u5f97\u5b57\u5178\u5e8f\u6700\u5c0f\u7684\u6392\u5e8f<\/li>\n\n\n\n
        4. \u52a8\u6001\u56fe\u5904\u7406<\/strong>\uff1aKahn\u7b97\u6cd5\u66f4\u9002\u5408\u52a8\u6001\u53d8\u5316\u7684\u56fe\u7ed3\u6784<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"
          \u62d3\u6251\u6392\u5e8f(Topological Sorting)\u662f\u9488\u5bf9\u6709\u5411\u65e0\u73af\u56fe(DAG, Directed Acyclic […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[4],"tags":[125],"class_list":["post-535","post","type-post","status-publish","format-standard","hentry","category-algorithm","tag-graph"],"_links":{"self":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=535"}],"version-history":[{"count":2,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/535\/revisions"}],"predecessor-version":[{"id":537,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/535\/revisions\/537"}],"wp:attachment":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}