{"id":85422,"date":"2026-07-27T14:32:26","date_gmt":"2026-07-27T06:32:26","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/85422.html"},"modified":"2026-07-27T14:32:26","modified_gmt":"2026-07-27T06:32:26","slug":"%e8%80%83%e7%a0%94408%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84-%e5%9b%be%e7%9a%84%e6%9c%80%e7%9f%ad%e8%b7%af%e5%be%84%e4%b8%8e%e6%9c%80%e5%b0%8f%e7%94%9f%e6%88%90%e6%a0%91%e7%ae%97","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/85422.html","title":{"rendered":"\u8003\u7814408\u6570\u636e\u7ed3\u6784\u2014\u2014\u56fe\u7684\u6700\u77ed\u8def\u5f84\u4e0e\u6700\u5c0f\u751f\u6210\u6811\u7b97\u6cd5\u5168\u89e3"},"content":{"rendered":"<p>\u672c\u6587\u7cfb\u7edf\u68b3\u7406408\u6570\u636e\u7ed3\u6784\u56fe\u8bba\u6838\u5fc3\u7b97\u6cd5&#xff0c;\u8986\u76d6Dijkstra\u3001Floyd\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u53caPrim\u3001Kruskal\u6700\u5c0f\u751f\u6210\u6811\u7b97\u6cd5&#xff0c;\u7ed3\u5408\u5386\u5e74\u771f\u9898\u51fa\u9898\u89d2\u5ea6\u8fdb\u884c\u6df1\u5ea6\u5206\u6790&#xff0c;\u9002\u5408\u8ba1\u7b97\u673a\u4e13\u4e1a\u8003\u7814\u540c\u5b66\u53c2\u8003\u3002<\/p>\n<h3>\u76ee\u5f55<\/h3>\n<ul>\n<li>\u4e00\u3001\u56fe\u7b97\u6cd5\u5728408\u8003\u8bd5\u4e2d\u7684\u5730\u4f4d<\/li>\n<li>\u4e8c\u3001\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\n<ul>\n<li>2.1 Dijkstra\u7b97\u6cd5<\/li>\n<li>2.2 Floyd\u7b97\u6cd5<\/li>\n<li>2.3 \u4e24\u79cd\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u5bf9\u6bd4<\/li>\n<\/ul>\n<\/li>\n<li>\u4e09\u3001\u6700\u5c0f\u751f\u6210\u6811\u7b97\u6cd5\n<ul>\n<li>3.1 Prim\u7b97\u6cd5<\/li>\n<li>3.2 Kruskal\u7b97\u6cd5<\/li>\n<li>3.3 Prim\u4e0eKruskal\u5bf9\u6bd4<\/li>\n<\/ul>\n<\/li>\n<li>\u56db\u3001\u7b97\u6cd5\u590d\u6742\u5ea6\u6c47\u603b\u5bf9\u6bd4<\/li>\n<li>\u4e94\u3001\u5386\u5e74\u771f\u9898\u8003\u67e5\u5206\u6790<\/li>\n<li>\u516d\u3001\u5b66\u4e60\u8d44\u6e90\u63a8\u8350<\/li>\n<\/ul>\n<hr \/>\n<h3>\u4e00\u3001\u56fe\u7b97\u6cd5\u5728408\u8003\u8bd5\u4e2d\u7684\u5730\u4f4d<\/h3>\n<p>408\u8ba1\u7b97\u673a\u5b66\u79d1\u4e13\u4e1a\u57fa\u7840\u7efc\u5408\u4e2d&#xff0c;\u6570\u636e\u7ed3\u6784\u7ea6\u536045\u5206\u6bd4\u91cd&#xff0c;\u800c\u56fe\u8bba\u76f8\u5173\u7b97\u6cd5\u51e0\u4e4e\u6bcf\u5e74\u5fc5\u8003&#xff0c;\u5e38\u51fa\u73b0\u5728\u9009\u62e9\u9898\u548c\u7b97\u6cd5\u5927\u9898\u4e2d\u3002\u8fd110\u5e74\u771f\u9898\u4e2d&#xff0c;\u56fe\u7b97\u6cd5\u76f8\u5173\u9898\u76ee\u51fa\u73b0\u9891\u7387\u5982\u4e0b&#xff1a;<\/p>\n<table>\n<tr>\u8003\u70b9\u51fa\u73b0\u5e74\u4efd\u9898\u578b<\/tr>\n<tbody>\n<tr>\n<td>Dijkstra\u7b97\u6cd5<\/td>\n<td>2015\u30012017\u30012019\u30012021\u30012023<\/td>\n<td>\u9009\u62e9\/\u7b80\u7b54<\/td>\n<\/tr>\n<tr>\n<td>Floyd\u7b97\u6cd5<\/td>\n<td>2016\u30012018\u30012022<\/td>\n<td>\u9009\u62e9<\/td>\n<\/tr>\n<tr>\n<td>Prim\u7b97\u6cd5<\/td>\n<td>2014\u30012017\u30012020\u30012023<\/td>\n<td>\u9009\u62e9\/\u586b\u7a7a<\/td>\n<\/tr>\n<tr>\n<td>Kruskal\u7b97\u6cd5<\/td>\n<td>2015\u30012019\u30012021\u30012022<\/td>\n<td>\u9009\u62e9\/\u586b\u7a7a<\/td>\n<\/tr>\n<tr>\n<td>\u7efc\u5408\u5e94\u7528<\/td>\n<td>2020\u30012023<\/td>\n<td>\u7b97\u6cd5\u8bbe\u8ba1\u9898<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u53ef\u4ee5\u770b\u51fa&#xff0c;\u6700\u77ed\u8def\u5f84\u548c\u6700\u5c0f\u751f\u6210\u6811\u662f\u9ad8\u9891\u8003\u70b9&#xff0c;\u9700\u8981\u6df1\u5165\u7406\u89e3\u7b97\u6cd5\u539f\u7406\u5e76\u80fd\u624b\u5199\u51fa\u6838\u5fc3\u4ee3\u7801\u3002<\/p>\n<hr \/>\n<h3>\u4e8c\u3001\u6700\u77ed\u8def\u5f84\u7b97\u6cd5<\/h3>\n<h4>2.1 Dijkstra\u7b97\u6cd5<\/h4>\n<p>\u6838\u5fc3\u601d\u60f3&#xff1a;\u8d2a\u5fc3\u7b56\u7565&#xff0c;\u4ece\u6e90\u70b9\u51fa\u53d1&#xff0c;\u6bcf\u6b21\u9009\u53d6\u8ddd\u79bb\u6700\u77ed\u7684\u5df2\u786e\u5b9a\u9876\u70b9&#xff0c;\u7528\u8be5\u9876\u70b9\u66f4\u65b0\u5176\u4f59\u9876\u70b9\u7684\u8ddd\u79bb\u4f30\u8ba1\u503c\u3002\u9002\u7528\u4e8e\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff0c;\u8981\u6c42\u56fe\u4e2d\u8fb9\u6743\u975e\u8d1f\u3002<\/p>\n<p>\u7b97\u6cd5\u6b65\u9aa4&#xff1a;<\/p>\n<li>\u521d\u59cb\u5316&#xff1a;\u5c06\u6e90\u70b9\u8ddd\u79bb\u8bbe\u4e3a0&#xff0c;\u5176\u4f59\u9876\u70b9\u8ddd\u79bb\u8bbe\u4e3a\u221e&#xff0c;\u6240\u6709\u9876\u70b9\u6807\u8bb0\u4e3a\u672a\u8bbf\u95ee<\/li>\n<li>\u4ece\u672a\u8bbf\u95ee\u9876\u70b9\u4e2d\u9009\u53d6\u8ddd\u79bb\u6700\u5c0f\u7684\u9876\u70b9u&#xff0c;\u6807\u8bb0\u4e3a\u5df2\u8bbf\u95ee<\/li>\n<li>\u5bf9u\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9v&#xff0c;\u82e5dist[u] &#043; weight(u,v) &lt; dist[v]&#xff0c;\u5219\u66f4\u65b0dist[v]<\/li>\n<li>\u91cd\u590d\u6b65\u9aa42-3&#xff0c;\u76f4\u5230\u6240\u6709\u9876\u70b9\u5747\u5df2\u8bbf\u95ee<\/li>\n<p>C\u8bed\u8a00\u5b9e\u73b0&#xff1a;<\/p>\n<p><span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">define<\/span> <span class=\"token macro-name\">MAXVEX<\/span> <span class=\"token expression\"><span class=\"token number\">100<\/span><\/span><\/span><br \/>\n<span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">define<\/span> <span class=\"token macro-name\">INFINITY<\/span> <span class=\"token expression\"><span class=\"token number\">65535<\/span><\/span><\/span><\/p>\n<p><span class=\"token keyword\">typedef<\/span> <span class=\"token keyword\">struct<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> vexs<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>          <span class=\"token comment\">\/\/ \u9876\u70b9\u8868<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> arc<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>   <span class=\"token comment\">\/\/ \u90bb\u63a5\u77e9\u9635<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> vexNum<span class=\"token punctuation\">,<\/span> arcNum<span class=\"token punctuation\">;<\/span>        <span class=\"token comment\">\/\/ \u9876\u70b9\u6570\u548c\u8fb9\u6570<\/span><br \/>\n<span class=\"token punctuation\">}<\/span> MGraph<span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">Dijkstra<\/span><span class=\"token punctuation\">(<\/span>MGraph G<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> v0<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> dist<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> path<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> final<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>  <span class=\"token comment\">\/\/ \u6807\u8bb0\u9876\u70b9\u662f\u5426\u5df2\u6c42\u5f97\u6700\u77ed\u8def\u5f84<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u521d\u59cb\u5316<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        final<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>v0<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> INFINITY<span class=\"token punctuation\">)<\/span><br \/>\n            path<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> v0<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">else<\/span><br \/>\n            path<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    final<span class=\"token punctuation\">[<\/span>v0<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    dist<span class=\"token punctuation\">[<\/span>v0<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u4e3b\u5faa\u73af<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        min <span class=\"token operator\">&#061;<\/span> INFINITY<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span>  <span class=\"token comment\">\/\/ \u627e\u6700\u5c0fdist<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>final<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&amp;&amp;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> min<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                k <span class=\"token operator\">&#061;<\/span> j<span class=\"token punctuation\">;<\/span><br \/>\n                min <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        final<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span>  <span class=\"token comment\">\/\/ \u6807\u8bb0\u5df2\u8bbf\u95ee<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u66f4\u65b0\u90bb\u63a5\u9876\u70b9\u8ddd\u79bb<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>final<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&amp;&amp;<\/span> min <span class=\"token operator\">&#043;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> min <span class=\"token operator\">&#043;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                path<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff1a;\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u5b58\u50a8\u65f6&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(V\u00b2)&#xff1b;\u82e5\u4f7f\u7528\u90bb\u63a5\u8868&#043;\u4f18\u5148\u961f\u5217&#xff08;\u5806\u4f18\u5316&#xff09;&#xff0c;\u53ef\u4f18\u5316\u81f3O((V&#043;E)logV)\u3002408\u8003\u8bd5\u4e2d\u5e38\u8003\u67e5\u90bb\u63a5\u77e9\u9635\u7248\u672c\u7684\u624b\u5199\u6a21\u62df\u3002<\/p>\n<hr \/>\n<h4>2.2 Floyd\u7b97\u6cd5<\/h4>\n<p>\u6838\u5fc3\u601d\u60f3&#xff1a;\u52a8\u6001\u89c4\u5212\u601d\u60f3&#xff0c;\u901a\u8fc7\u9010\u6b65\u63d2\u5165\u4e2d\u95f4\u9876\u70b9\u6765\u66f4\u65b0\u4efb\u610f\u4e24\u70b9\u95f4\u7684\u6700\u77ed\u8ddd\u79bb\u3002\u9002\u7528\u4e8e\u591a\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff0c;\u53ef\u5904\u7406\u8d1f\u6743\u8fb9&#xff08;\u4f46\u4e0d\u80fd\u6709\u8d1f\u6743\u56de\u8def&#xff09;\u3002<\/p>\n<p>\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b&#xff1a;<\/p>\n<p>D^(k)[i][j] &#061; min(D^(k-1)[i][j], D^(k-1)[i][k] &#043; D^(k-1)[k][j])<\/p>\n<p>\u5176\u4e2dk\u4e3a\u5f53\u524d\u5141\u8bb8\u7ecf\u8fc7\u7684\u4e2d\u95f4\u9876\u70b9\u7f16\u53f7\u3002<\/p>\n<p>C\u8bed\u8a00\u5b9e\u73b0&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">Floyd<\/span><span class=\"token punctuation\">(<\/span>MGraph G<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> dist<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> path<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u521d\u59cb\u5316\u8ddd\u79bb\u77e9\u9635\u548c\u8def\u5f84\u77e9\u9635<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">!&#061;<\/span> j <span class=\"token operator\">&amp;&amp;<\/span> dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> INFINITY<span class=\"token punctuation\">)<\/span><br \/>\n                path<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">else<\/span><br \/>\n                path<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u4e09\u91cd\u5faa\u73af&#xff1a;k\u4e3a\u4e2d\u95f4\u9876\u70b9<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> k<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> dist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> dist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    path<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> path<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u6ce8\u610f&#xff1a;\u4e09\u91cd\u5faa\u73af\u7684\u5d4c\u5957\u987a\u5e8f\u4e0d\u80fd\u98a0\u5012&#xff0c;\u6700\u5916\u5c42\u5fc5\u987b\u662fk&#xff08;\u4e2d\u95f4\u9876\u70b9&#xff09;&#xff0c;\u8fd9\u662fFloyd\u7b97\u6cd5\u6b63\u786e\u6027\u7684\u5173\u952e\u3002<\/p>\n<hr \/>\n<h4>2.3 \u4e24\u79cd\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u5bf9\u6bd4<\/h4>\n<table>\n<tr>\u5bf9\u6bd4\u7ef4\u5ea6DijkstraFloyd<\/tr>\n<tbody>\n<tr>\n<td>\u9002\u7528\u573a\u666f<\/td>\n<td>\u5355\u6e90\u6700\u77ed\u8def\u5f84<\/td>\n<td>\u591a\u6e90&#xff08;\u5168\u6e90&#xff09;\u6700\u77ed\u8def\u5f84<\/td>\n<\/tr>\n<tr>\n<td>\u8fb9\u6743\u9650\u5236<\/td>\n<td>\u975e\u8d1f\u6743<\/td>\n<td>\u53ef\u5904\u7406\u8d1f\u6743&#xff08;\u65e0\u8d1f\u6743\u56de\u8def&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u65f6\u95f4\u590d\u6742\u5ea6<\/td>\n<td>O(V\u00b2) \/ O((V&#043;E)logV)\u5806\u4f18\u5316<\/td>\n<td>O(V\u00b3)<\/td>\n<\/tr>\n<tr>\n<td>\u7a7a\u95f4\u590d\u6742\u5ea6<\/td>\n<td>O(V)<\/td>\n<td>O(V\u00b2)<\/td>\n<\/tr>\n<tr>\n<td>\u7b97\u6cd5\u601d\u60f3<\/td>\n<td>\u8d2a\u5fc3<\/td>\n<td>\u52a8\u6001\u89c4\u5212<\/td>\n<\/tr>\n<tr>\n<td>\u5b58\u50a8\u7ed3\u6784<\/td>\n<td>\u90bb\u63a5\u77e9\u9635\/\u90bb\u63a5\u8868<\/td>\n<td>\u90bb\u63a5\u77e9\u9635<\/td>\n<\/tr>\n<tr>\n<td>408\u8003\u67e5\u9891\u7387<\/td>\n<td>\u2605\u2605\u2605\u2605\u2605<\/td>\n<td>\u2605\u2605\u2605\u2605<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u4e09\u3001\u6700\u5c0f\u751f\u6210\u6811\u7b97\u6cd5<\/h3>\n<p>\u6700\u5c0f\u751f\u6210\u6811&#xff08;MST&#xff09;\u7684\u76ee\u6807&#xff1a;\u5728\u8fde\u901a\u56fe\u4e2d\u627e\u5230\u4e00\u68f5\u5305\u542b\u6240\u6709\u9876\u70b9\u7684\u751f\u6210\u6811&#xff0c;\u4f7f\u5f97\u6811\u4e0a\u6240\u6709\u8fb9\u7684\u6743\u503c\u4e4b\u548c\u6700\u5c0f\u3002<\/p>\n<h4>3.1 Prim\u7b97\u6cd5<\/h4>\n<p>\u6838\u5fc3\u601d\u60f3&#xff1a;\u4ece\u67d0\u4e00\u9876\u70b9\u5f00\u59cb&#xff0c;\u9010\u6b65\u6269\u5c55\u751f\u6210\u6811\u3002\u6bcf\u6b21\u4ece\u672a\u52a0\u5165\u6811\u4e2d\u7684\u9876\u70b9\u4e2d&#xff0c;\u9009\u53d6\u4e0e\u5f53\u524d\u6811\u76f8\u8fde\u7684\u8fb9\u6743\u6700\u5c0f\u7684\u9876\u70b9\u52a0\u5165\u3002\u5c5e\u4e8e\u52a0\u70b9\u6cd5\u3002<\/p>\n<p>C\u8bed\u8a00\u6838\u5fc3\u903b\u8f91&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">Prim<\/span><span class=\"token punctuation\">(<\/span>MGraph G<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> closedge<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> lowcost<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>   <span class=\"token comment\">\/\/ \u8bb0\u5f55\u751f\u6210\u6811\u5230\u5404\u9876\u70b9\u7684\u6700\u5c0f\u8fb9\u6743<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> adjvex<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u8bb0\u5f55\u6700\u5c0f\u8fb9\u5bf9\u5e94\u7684\u6811\u4e2d\u9876\u70b9<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u4ece\u9876\u70b90\u5f00\u59cb\u6784\u9020\u6700\u5c0f\u751f\u6210\u6811<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        lowcost<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        adjvex<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    lowcost<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span>  <span class=\"token comment\">\/\/ \u9876\u70b90\u52a0\u5165\u751f\u6210\u6811<\/span><\/p>\n<p>    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u627elowcost\u4e2d\u6700\u5c0f\u503c<\/span><br \/>\n        min <span class=\"token operator\">&#061;<\/span> INFINITY<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> <span class=\"token number\">0<\/span> <span class=\"token operator\">&amp;&amp;<\/span> lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> min<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                min <span class=\"token operator\">&#061;<\/span> lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                k <span class=\"token operator\">&#061;<\/span> j<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u8f93\u51fa\u8fb9(adjvex[k], k)<\/span><br \/>\n        <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;(%d, %d)&#034;<\/span><span class=\"token punctuation\">,<\/span> adjvex<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        lowcost<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span>  <span class=\"token comment\">\/\/ \u9876\u70b9k\u52a0\u5165\u751f\u6210\u6811<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u66f4\u65b0lowcost<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>vexNum<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> <span class=\"token number\">0<\/span> <span class=\"token operator\">&amp;&amp;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                lowcost<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> G<span class=\"token punctuation\">.<\/span>arc<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                adjvex<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<hr \/>\n<h4>3.2 Kruskal\u7b97\u6cd5<\/h4>\n<p>\u6838\u5fc3\u601d\u60f3&#xff1a;\u5c06\u6240\u6709\u8fb9\u6309\u6743\u503c\u4ece\u5c0f\u5230\u5927\u6392\u5e8f&#xff0c;\u4f9d\u6b21\u9009\u53d6\u4e0d\u6784\u6210\u56de\u8def\u7684\u8fb9\u52a0\u5165\u751f\u6210\u6811\u3002\u5c5e\u4e8e\u52a0\u8fb9\u6cd5&#xff0c;\u9700\u8981\u501f\u52a9\u5e76\u67e5\u96c6\u6765\u5224\u65ad\u662f\u5426\u6784\u6210\u56de\u8def\u3002<\/p>\n<p>C\u8bed\u8a00\u6838\u5fc3\u903b\u8f91&#xff1a;<\/p>\n<p><span class=\"token keyword\">typedef<\/span> <span class=\"token keyword\">struct<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> begin<span class=\"token punctuation\">,<\/span> end<span class=\"token punctuation\">,<\/span> weight<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span> Edge<span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token comment\">\/\/ \u5e76\u67e5\u96c6\u67e5\u627e\u6839\u8282\u70b9<\/span><br \/>\n<span class=\"token keyword\">int<\/span> <span class=\"token function\">Find<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> parent<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> f<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>parent<span class=\"token punctuation\">[<\/span>f<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        f <span class=\"token operator\">&#061;<\/span> parent<span class=\"token punctuation\">[<\/span>f<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> f<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">Kruskal<\/span><span class=\"token punctuation\">(<\/span>MGraph G<span class=\"token punctuation\">,<\/span> Edge edges<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> parent<span class=\"token punctuation\">[<\/span>MAXVEX<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> n<span class=\"token punctuation\">,<\/span> m<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u6309weight\u6392\u5e8f\u540e\u4f9d\u6b21\u5904\u7406\u6bcf\u6761\u8fb9<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> G<span class=\"token punctuation\">.<\/span>arcNum<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        n <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">Find<\/span><span class=\"token punctuation\">(<\/span>parent<span class=\"token punctuation\">,<\/span> edges<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>begin<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        m <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">Find<\/span><span class=\"token punctuation\">(<\/span>parent<span class=\"token punctuation\">,<\/span> edges<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>end<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">!&#061;<\/span> m<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span>  <span class=\"token comment\">\/\/ \u4e0d\u5728\u540c\u4e00\u96c6\u5408&#xff0c;\u4e0d\u6784\u6210\u56de\u8def<\/span><br \/>\n            parent<span class=\"token punctuation\">[<\/span>n<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> m<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;(%d, %d) weight&#061;%d\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n                   edges<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>begin<span class=\"token punctuation\">,<\/span> edges<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>end<span class=\"token punctuation\">,<\/span> edges<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>weight<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<hr \/>\n<h4>3.3 Prim\u4e0eKruskal\u5bf9\u6bd4<\/h4>\n<table>\n<tr>\u5bf9\u6bd4\u7ef4\u5ea6PrimKruskal<\/tr>\n<tbody>\n<tr>\n<td>\u7b56\u7565<\/td>\n<td>\u52a0\u70b9\u6cd5<\/td>\n<td>\u52a0\u8fb9\u6cd5<\/td>\n<\/tr>\n<tr>\n<td>\u9002\u7528\u573a\u666f<\/td>\n<td>\u7a20\u5bc6\u56fe&#xff08;\u8fb9\u591a&#xff09;<\/td>\n<td>\u7a00\u758f\u56fe&#xff08;\u8fb9\u5c11&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u65f6\u95f4\u590d\u6742\u5ea6<\/td>\n<td>O(V\u00b2)&#xff08;\u90bb\u63a5\u77e9\u9635&#xff09;<\/td>\n<td>O(ElogE)&#xff08;\u8fb9\u6392\u5e8f&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u8f85\u52a9\u7ed3\u6784<\/td>\n<td>lowcost\u6570\u7ec4<\/td>\n<td>\u5e76\u67e5\u96c6<\/td>\n<\/tr>\n<tr>\n<td>\u5b9e\u73b0\u96be\u5ea6<\/td>\n<td>\u4e2d\u7b49<\/td>\n<td>\u9700\u5b9e\u73b0\u6392\u5e8f&#043;\u5e76\u67e5\u96c6<\/td>\n<\/tr>\n<tr>\n<td>408\u8003\u67e5\u91cd\u70b9<\/td>\n<td>\u624b\u52a8\u6a21\u62df\u9009\u70b9\u8fc7\u7a0b<\/td>\n<td>\u624b\u52a8\u6a21\u62df\u9009\u8fb9&#043;\u5224\u65ad\u56de\u8def<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u56db\u3001\u7b97\u6cd5\u590d\u6742\u5ea6\u6c47\u603b\u5bf9\u6bd4<\/h3>\n<table>\n<tr>\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u7a7a\u95f4\u590d\u6742\u5ea6\u9002\u7528\u56fe\u7c7b\u578b<\/tr>\n<tbody>\n<tr>\n<td>Dijkstra&#xff08;\u90bb\u63a5\u77e9\u9635&#xff09;<\/td>\n<td>O(V\u00b2)<\/td>\n<td>O(V)<\/td>\n<td>\u7a20\u5bc6\u56fe<\/td>\n<\/tr>\n<tr>\n<td>Dijkstra&#xff08;\u5806\u4f18\u5316&#xff09;<\/td>\n<td>O((V&#043;E)logV)<\/td>\n<td>O(V&#043;E)<\/td>\n<td>\u7a00\u758f\u56fe<\/td>\n<\/tr>\n<tr>\n<td>Floyd<\/td>\n<td>O(V\u00b3)<\/td>\n<td>O(V\u00b2)<\/td>\n<td>\u5168\u6e90\u3001\u4efb\u610f\u5bc6\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>Prim&#xff08;\u90bb\u63a5\u77e9\u9635&#xff09;<\/td>\n<td>O(V\u00b2)<\/td>\n<td>O(V)<\/td>\n<td>\u7a20\u5bc6\u56fe<\/td>\n<\/tr>\n<tr>\n<td>Kruskal<\/td>\n<td>O(ElogE)<\/td>\n<td>O(E)<\/td>\n<td>\u7a00\u758f\u56fe<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u5907\u8003\u63d0\u793a&#xff1a;408\u8003\u8bd5\u4e2d\u5e38\u8981\u6c42\u8003\u751f\u6839\u636e\u5177\u4f53\u56fe\u7ed3\u6784\u624b\u52a8\u6a21\u62df\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b&#xff0c;\u8bb0\u5f55\u6bcf\u8f6e\u9009\u62e9\u7ed3\u679c\u3002\u5efa\u8bae\u5bf9\u6bcf\u4e2a\u7b97\u6cd5\u81f3\u5c11\u624b\u7b972-3\u9053\u4e0d\u540c\u89c4\u6a21\u7684\u9898\u76ee\u3002<\/p>\n<hr \/>\n<h3>\u4e94\u3001\u5386\u5e74\u771f\u9898\u8003\u67e5\u5206\u6790<\/h3>\n<p>\u6839\u636e\u5bf9\u8fd110\u5e74408\u771f\u9898\u7684\u7edf\u8ba1&#xff0c;\u56fe\u7b97\u6cd5\u51fa\u9898\u89d2\u5ea6\u4e3b\u8981\u5305\u62ec&#xff1a;<\/p>\n<p>1. \u7b97\u6cd5\u8fc7\u7a0b\u6a21\u62df\u9898&#xff08;\u9ad8\u9891&#xff09;<\/p>\n<p>\u7ed9\u51fa\u4e00\u5f20\u5e26\u6743\u56fe&#xff0c;\u8981\u6c42\u6309\u7167Dijkstra\/Prim\/Kruskal\u7684\u6b65\u9aa4\u9010\u6b65\u5199\u51fa\u6267\u884c\u8fc7\u7a0b&#xff0c;\u8bb0\u5f55\u6bcf\u8f6e\u9009\u53d6\u7684\u9876\u70b9\u548c\u66f4\u65b0\u540e\u7684\u8ddd\u79bb\u6570\u7ec4\u3002<\/p>\n<p>2. \u7b97\u6cd5\u6027\u8d28\u5224\u65ad\u9898&#xff08;\u4e2d\u9891&#xff09;<\/p>\n<p>\u4f8b\u5982&#xff1a;<\/p>\n<ul>\n<li>Dijkstra\u80fd\u5426\u5904\u7406\u8d1f\u6743\u8fb9&#xff1f;\u4e3a\u4ec0\u4e48&#xff1f;<\/li>\n<li>Floyd\u7b97\u6cd5\u4e2d\u4e09\u91cd\u5faa\u73af\u987a\u5e8f\u80fd\u5426\u6539\u53d8&#xff1f;<\/li>\n<li>Prim\u7b97\u6cd5\u548cDijkstra\u7b97\u6cd5\u7684\u5f02\u540c\u70b9\u662f\u4ec0\u4e48&#xff1f;<\/li>\n<\/ul>\n<p>3. \u4ee3\u7801\u586b\u7a7a\u9898&#xff08;\u4e2d\u9891&#xff09;<\/p>\n<p>\u7ed9\u51fa\u4e0d\u5b8c\u6574\u7684\u7b97\u6cd5\u4ee3\u7801&#xff0c;\u8981\u6c42\u8865\u5168\u5173\u952e\u903b\u8f91&#xff0c;\u5982\u66f4\u65b0\u8ddd\u79bb\u3001\u6807\u8bb0\u8bbf\u95ee\u72b6\u6001\u7b49\u3002<\/p>\n<p>4. \u7b97\u6cd5\u8bbe\u8ba1\u9898&#xff08;\u4f4e\u9891\u4f46\u5206\u503c\u9ad8&#xff09;<\/p>\n<p>\u7ed3\u5408\u5177\u4f53\u5e94\u7528\u573a\u666f&#xff08;\u5982\u4ea4\u901a\u7f51\u7edc\u3001\u901a\u4fe1\u7f51\u7edc&#xff09;&#xff0c;\u8981\u6c42\u8bbe\u8ba1\u7b97\u6cd5\u6c42\u6700\u77ed\u8def\u5f84\u6216\u6700\u5c0f\u4ee3\u4ef7&#xff0c;\u5e76\u5206\u6790\u590d\u6742\u5ea6\u3002<\/p>\n<table>\n<tr>\u5e74\u4efd\u9898\u53f7\u8003\u67e5\u5185\u5bb9\u5206\u503c<\/tr>\n<tbody>\n<tr>\n<td>2023<\/td>\n<td>36-37<\/td>\n<td>Prim\u7b97\u6cd5\u8fc7\u7a0b\u6a21\u62df<\/td>\n<td>8\u5206<\/td>\n<\/tr>\n<tr>\n<td>2023<\/td>\n<td>41<\/td>\n<td>\u6700\u77ed\u8def\u5f84\u5e94\u7528\u8bbe\u8ba1<\/td>\n<td>10\u5206<\/td>\n<\/tr>\n<tr>\n<td>2022<\/td>\n<td>9-10<\/td>\n<td>Floyd\u7b97\u6cd5\u6027\u8d28<\/td>\n<td>4\u5206<\/td>\n<\/tr>\n<tr>\n<td>2022<\/td>\n<td>35<\/td>\n<td>Kruskal\u7b97\u6cd5\u6a21\u62df<\/td>\n<td>8\u5206<\/td>\n<\/tr>\n<tr>\n<td>2021<\/td>\n<td>7-8<\/td>\n<td>Dijkstra\u6267\u884c\u8fc7\u7a0b<\/td>\n<td>4\u5206<\/td>\n<\/tr>\n<tr>\n<td>2021<\/td>\n<td>42<\/td>\n<td>\u56fe\u7efc\u5408\u5e94\u7528<\/td>\n<td>10\u5206<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr 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