{"id":532,"date":"2020-08-12T23:22:38","date_gmt":"2020-08-12T15:22:38","guid":{"rendered":"http:\/\/120.55.184.7\/?p=532"},"modified":"2025-04-15T18:15:16","modified_gmt":"2025-04-15T10:15:16","slug":"%e6%9c%80%e5%b0%8f%e7%94%9f%e6%88%90%e6%a0%91mst","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=532","title":{"rendered":"\u6700\u5c0f\u751f\u6210\u6811(MST)"},"content":{"rendered":"\n<p>\u6700\u5c0f\u751f\u6210\u6811(Minimum Spanning Tree, MST)\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u6307\u5728\u4e00\u4e2a<strong>\u8fde\u901a\u52a0\u6743\u65e0\u5411\u56fe<\/strong>\u4e2d\uff0c\u627e\u5230\u4e00\u68f5\u5305\u542b\u6240\u6709\u9876\u70b9\u7684\u751f\u6210\u6811\uff0c\u4e14\u8fd9\u68f5\u6811\u6240\u6709\u8fb9\u7684\u6743\u503c\u4e4b\u548c\u6700\u5c0f\u3002\u6700\u5c0f\u751f\u6210\u6811\u5728\u5b9e\u9645\u4e2d\u6709\u5e7f\u6cdb\u5e94\u7528\uff0c\u5982\u7f51\u7edc\u8bbe\u8ba1\u3001\u7535\u8def\u5e03\u7ebf\u3001\u4ea4\u901a\u89c4\u5212\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e00\u3001\u6700\u5c0f\u751f\u6210\u6811\u7684\u57fa\u672c\u6027\u8d28<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5b9a\u4e49<\/strong>\uff1a\u5bf9\u4e8e\u8fde\u901a\u52a0\u6743\u65e0\u5411\u56feG=(V,E)\uff0c\u6700\u5c0f\u751f\u6210\u6811T\u662fG\u7684\u4e00\u4e2a\u5b50\u56fe\uff0c\u6ee1\u8db3\uff1a<br>\u2022 \u5305\u542bG\u7684\u6240\u6709\u9876\u70b9<br>\u2022 \u662f\u4e00\u68f5\u6811\uff08\u65e0\u73af\u4e14\u8fde\u901a\uff09<br>\u2022 \u6240\u6709\u8fb9\u7684\u6743\u503c\u4e4b\u548c\u6700\u5c0f<\/li>\n\n\n\n<li><strong>\u6027\u8d28<\/strong>\uff1a<br>\u2022 <strong>\u73af\u5207\u5206\u6027\u8d28<\/strong>\uff1a\u56fe\u4e2d\u4efb\u610f\u73af\uff0c\u5220\u9664\u6743\u503c\u6700\u5927\u7684\u8fb9\u540e\uff0c\u5269\u4f59\u8fb9\u4ecd\u6784\u6210\u751f\u6210\u6811<br>\u2022 <strong>\u8fb9\u9009\u62e9\u6027\u8d28<\/strong>\uff1a\u82e5\u8fb9\u662f\u8fde\u63a5\u4e24\u4e2a\u5b50\u6811\u7684\u6700\u5c0f\u6743\u8fb9\uff0c\u5219\u8be5\u8fb9\u5fc5\u5c5e\u4e8e\u67d0\u68f5MST<br>\u2022 <strong>\u552f\u4e00\u6027<\/strong>\uff1a\u5f53\u6240\u6709\u8fb9\u6743\u503c\u4e0d\u76f8\u540c\u65f6\uff0cMST\u552f\u4e00\uff1b\u5426\u5219\u53ef\u80fd\u5b58\u5728\u591a\u4e2aMST<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e8c\u3001\u7ecf\u5178\u7b97\u6cd5<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. Prim\u7b97\u6cd5\uff08\u666e\u91cc\u59c6\u7b97\u6cd5\uff09<\/h3>\n\n\n\n<p><strong>\u57fa\u672c\u601d\u60f3<\/strong>\uff1a\u4ece\u9876\u70b9\u51fa\u53d1\uff0c\u9010\u6b65\u6269\u5c55\u751f\u6210\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u8fde\u63a5\u751f\u6210\u6811\u4e0e\u975e\u751f\u6210\u6811\u9876\u70b9\u7684\u6700\u5c0f\u6743\u8fb9\u3002<\/p>\n\n\n\n<p><strong>\u7b97\u6cd5\u6b65\u9aa4<\/strong>\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4efb\u9009\u4e00\u4e2a\u8d77\u59cb\u9876\u70b9\u52a0\u5165\u751f\u6210\u6811<\/li>\n\n\n\n<li>\u5728\u6240\u6709\u8fde\u63a5\u751f\u6210\u6811\u4e0e\u975e\u751f\u6210\u6811\u9876\u70b9\u7684\u8fb9\u4e2d\uff0c\u9009\u62e9\u6743\u503c\u6700\u5c0f\u7684\u8fb9\u52a0\u5165\u751f\u6210\u6811<\/li>\n\n\n\n<li>\u5c06\u65b0\u9876\u70b9\u52a0\u5165\u751f\u6210\u6811\u96c6\u5408<\/li>\n\n\n\n<li>\u91cd\u590d\u6b65\u9aa42-3\uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u90fd\u52a0\u5165\u751f\u6210\u6811<\/li>\n<\/ol>\n\n\n\n<p><strong>\u65f6\u95f4\u590d\u6742\u5ea6<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\uff1aO(V\u00b2)<\/li>\n\n\n\n<li>\u4e8c\u53c9\u5806\u4f18\u5316\uff1aO(E log V)<\/li>\n\n\n\n<li>\u6590\u6ce2\u90a3\u5951\u5806\u4f18\u5316\uff1aO(E + V log V)<\/li>\n<\/ul>\n\n\n\n<p><strong>C++\u5b9e\u73b0<\/strong>\uff1a<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\nint prim(vector&lt;vector&lt;pair&lt;int, int&gt;&gt;&gt;&amp; graph, int start) {\n    int n = graph.size();\n    vector&lt;int&gt; dist(n, INT_MAX);\n    vector&lt;bool&gt; visited(n, false);\n    priority_queue&lt;pair&lt;int, int&gt;, vector&lt;pair&lt;int, int&gt;&gt;, greater&lt;&gt;&gt; pq;\n\n    dist&#x5B;start] = 0;\n    pq.push({0, start});\n    int total = 0;\n\n    while (!pq.empty()) {\n        auto &#x5B;d, u] = pq.top(); pq.pop();\n        if (visited&#x5B;u]) continue;\n        visited&#x5B;u] = true;\n        total += d;\n\n        for (auto &#x5B;v, w] : graph&#x5B;u]) {\n            if (!visited&#x5B;v] &amp;&amp; w &lt; dist&#x5B;v]) {\n                dist&#x5B;v] = w;\n                pq.push({w, v});\n            }\n        }\n    }\n    return total;\n}\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">2. Kruskal\u7b97\u6cd5\uff08\u514b\u9c81\u65af\u5361\u5c14\u7b97\u6cd5\uff09<\/h3>\n\n\n\n<p><strong>\u57fa\u672c\u601d\u60f3<\/strong>\uff1a\u6309\u8fb9\u6743\u503c\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff0c\u4f9d\u6b21\u9009\u62e9\u4e0d\u5f62\u6210\u73af\u7684\u8fb9\u52a0\u5165\u751f\u6210\u6811\u3002<\/p>\n\n\n\n<p><strong>\u7b97\u6cd5\u6b65\u9aa4<\/strong>\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5c06\u6240\u6709\u8fb9\u6309\u6743\u503c\u4ece\u5c0f\u5230\u5927\u6392\u5e8f<\/li>\n\n\n\n<li>\u521d\u59cb\u5316\u5e76\u67e5\u96c6\u6570\u636e\u7ed3\u6784<\/li>\n\n\n\n<li>\u4f9d\u6b21\u68c0\u67e5\u6bcf\u6761\u8fb9\uff0c\u5982\u679c\u8fb9\u7684\u4e24\u4e2a\u7aef\u70b9\u4e0d\u5728\u540c\u4e00\u96c6\u5408\uff0c\u5219\u52a0\u5165\u751f\u6210\u6811\u5e76\u5408\u5e76\u96c6\u5408<\/li>\n\n\n\n<li>\u91cd\u590d\u6b65\u9aa43\u76f4\u5230\u751f\u6210\u6811\u5305\u542bV-1\u6761\u8fb9<\/li>\n<\/ol>\n\n\n\n<p><strong>\u65f6\u95f4\u590d\u6742\u5ea6<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6392\u5e8f\u4e3b\u5bfc\uff1aO(E log E)<\/li>\n\n\n\n<li>\u5e76\u67e5\u96c6\u64cd\u4f5c\uff1a\u8fd1\u4f3cO(\u03b1(V))<\/li>\n\n\n\n<li>\u603b\u590d\u6742\u5ea6\uff1aO(E log E)<\/li>\n<\/ul>\n\n\n\n<p><strong>C++\u5b9e\u73b0<\/strong>\uff1a<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\nstruct DSU {\n    vector&lt;int&gt; parent;\n    DSU(int n) : parent(n) { iota(parent.begin(), parent.end(), 0); }\n    int find(int x) { return parent&#x5B;x] == x ? x : parent&#x5B;x] = find(parent&#x5B;x]); }\n    bool unite(int x, int y) {\n        x = find(x), y = find(y);\n        if (x == y) return false;\n        parent&#x5B;y] = x;\n        return true;\n    }\n};\n\nint kruskal(vector&lt;vector&lt;int&gt;&gt;&amp; edges, int n) {\n    sort(edges.begin(), edges.end(), &#x5B;](auto&amp; a, auto&amp; b) { return a&#x5B;2] &lt; b&#x5B;2]; });\n    DSU dsu(n);\n    int total = 0, count = 0;\n\n    for (auto&amp; e : edges) {\n        if (dsu.unite(e&#x5B;0], e&#x5B;1])) {\n            total += e&#x5B;2];\n            if (++count == n - 1) break;\n        }\n    }\n    return count == n - 1 ? total : -1;\n}\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u4e09\u3001\u7b97\u6cd5\u6bd4\u8f83\u4e0e\u9009\u62e9<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\u7279\u6027<\/th><th>Prim\u7b97\u6cd5<\/th><th>Kruskal\u7b97\u6cd5<\/th><\/tr><\/thead><tbody><tr><td><strong>\u601d\u60f3<\/strong><\/td><td>\u9876\u70b9\u6269\u5c55<\/td><td>\u8fb9\u9009\u62e9<\/td><\/tr><tr><td><strong>\u6570\u636e\u7ed3\u6784<\/strong><\/td><td>\u4f18\u5148\u961f\u5217<\/td><td>\u5e76\u67e5\u96c6+\u6392\u5e8f<\/td><\/tr><tr><td><strong>\u65f6\u95f4\u590d\u6742\u5ea6<\/strong><\/td><td>O(E log V)<\/td><td>O(E log E)<\/td><\/tr><tr><td><strong>\u9002\u7528\u573a\u666f<\/strong><\/td><td>\u7a20\u5bc6\u56fe<\/td><td>\u7a00\u758f\u56fe<\/td><\/tr><tr><td><strong>\u5b9e\u73b0\u96be\u5ea6<\/strong><\/td><td>\u4e2d\u7b49<\/td><td>\u8f83\u7b80\u5355<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>\u9009\u62e9\u5efa\u8bae<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5f53\u56fe\u8f83\u7a20\u5bc6\uff08E\u2248V\u00b2\uff09\u65f6\uff0cPrim\u7b97\u6cd5\u66f4\u9ad8\u6548<\/li>\n\n\n\n<li>\u5f53\u56fe\u8f83\u7a00\u758f\uff08E\u2248V\uff09\u65f6\uff0cKruskal\u7b97\u6cd5\u66f4\u5408\u9002<\/li>\n\n\n\n<li>\u9700\u8981\u52a8\u6001\u7ef4\u62a4\u751f\u6210\u6811\u65f6\uff0cPrim\u7b97\u6cd5\u66f4\u65b9\u4fbf<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u56db\u3001\u5e94\u7528\u5b9e\u4f8b<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u7f51\u7edc\u8bbe\u8ba1<\/strong>\uff1a\u5728\u591a\u4e2a\u57ce\u5e02\u95f4\u5efa\u7acb\u901a\u4fe1\u7f51\u7edc\uff0c\u6700\u5c0f\u5316\u5e03\u7ebf\u6210\u672c<\/li>\n\n\n\n<li><strong>\u7535\u8def\u8bbe\u8ba1<\/strong>\uff1a\u8fde\u63a5\u591a\u4e2a\u5143\u4ef6\uff0c\u6700\u5c0f\u5316\u5bfc\u7ebf\u603b\u957f\u5ea6<\/li>\n\n\n\n<li><strong>\u805a\u7c7b\u5206\u6790<\/strong>\uff1a\u57fa\u4e8e\u76f8\u4f3c\u5ea6\u6784\u5efa\u5c42\u6b21\u805a\u7c7b\u7ed3\u6784<\/li>\n\n\n\n<li><strong>\u56fe\u50cf\u5206\u5272<\/strong>\uff1a\u5c06\u56fe\u50cf\u50cf\u7d20\u8fde\u63a5\u6210\u533a\u57df<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u5c0f\u751f\u6210\u6811(Minimum Spanning Tree, MST)\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u6307\u5728\u4e00\u4e2a\u8fde\u901a\u52a0\u6743\u65e0\u5411 [&hellip;]<\/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 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