{"id":54761,"date":"2025-08-13T11:00:28","date_gmt":"2025-08-13T03:00:28","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/54761.html"},"modified":"2025-08-13T11:00:28","modified_gmt":"2025-08-13T03:00:28","slug":"floyd%e7%ae%97%e6%b3%95%ef%bc%9a%e4%b8%89%e8%a1%8c%e4%bb%a3%e7%a0%81%e8%a7%a3%e5%86%b3%e6%9c%80%e7%9f%ad%e8%b7%af%e5%be%84","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/54761.html","title":{"rendered":"Floyd\u7b97\u6cd5\uff1a\u4e09\u884c\u4ee3\u7801\u89e3\u51b3\u6700\u77ed\u8def\u5f84"},"content":{"rendered":"<p>\u00a0 \u00a0 \u00a0 \u00a0 \u54c8\u55bd\u5927\u5bb6\u597d&#xff01;\u4eca\u5929\u54b1\u4eec\u6765\u804a\u804a\u56fe\u8bba\u91cc\u8d85\u7ecf\u5178\u7684 Floyd \u7b97\u6cd5 \u2014\u2014 \u8fd9\u4e2a\u80fd\u4e00\u6b21\u6027\u7b97\u51fa\u6240\u6709\u8282\u70b9\u4e4b\u95f4\u6700\u77ed\u8def\u5f84\u7684\u795e\u5947\u7b97\u6cd5\u3002\u4e0d\u7ba1\u4f60\u662f\u7b97\u6cd5\u65b0\u624b\u8fd8\u662f\u60f3\u590d\u4e60\u7684\u5907\u8003\u751f&#xff0c;\u8ddf\u7740\u6211\u4e00\u6b65\u4e00\u6b65\u6572\u4ee3\u7801&#xff0c;\u4fdd\u8bc1\u4f60\u5f7b\u5e95\u641e\u61c2\u5b83&#xff01;<\/p>\n<h3>\u4e00\u3001\u5148\u641e\u61c2&#xff1a;Floyd \u7b97\u6cd5\u5230\u5e95\u80fd\u5e72\u5565&#xff1f;<\/h3>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5728\u5f00\u59cb\u5199\u4ee3\u7801\u524d&#xff0c;\u54b1\u4eec\u5f97\u5148\u660e\u786e\u95ee\u9898&#xff1a;\u4ec0\u4e48\u662f\u591a\u6e90\u6700\u77ed\u8def\u5f84&#xff1f;<br \/>\n\u7b80\u5355\u8bf4\u5c31\u662f&#xff1a;\u7ed9\u4f60\u4e00\u5f20\u5e26\u6743\u56fe&#xff08;\u53ef\u4ee5\u6709\u5411\u4e5f\u53ef\u4ee5\u65e0\u5411&#xff09;&#xff0c;\u8981\u4f60\u6c42\u51fa\u4efb\u610f\u4e24\u4e2a\u8282\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n<p>\u6bd4\u5982\u8fd9\u6837\u4e00\u5f20\u56fe:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" alt=\"\" class=\"left\" height=\"220\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250813030027-689bffcb1ed11.png\" width=\"104\" \/><\/p>\n<p>\u6211\u4eec\u9700\u8981\u77e5\u9053 A \u5230 B\u3001A \u5230 C\u3001B \u5230 D&#8230; \u6240\u6709\u6210\u5bf9\u8282\u70b9\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u3002<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Floyd \u7b97\u6cd5\u7684\u4f18\u52bf\u5c31\u5728\u4e8e&#xff1a;\u7528\u6781\u5176\u7b80\u6d01\u7684\u4ee3\u7801\u5b9e\u73b0\u591a\u6e90\u6700\u77ed\u8def\u5f84\u8ba1\u7b97&#xff0c;\u6838\u5fc3\u903b\u8f91\u751a\u81f3\u80fd\u6d53\u7f29\u6210\u4e09\u91cd\u5faa\u73af&#xff01;&#xff08;\u5c31\u662f\u5bb9\u6613\u65f6\u95f4\u8d85\u9650&#xff09;<\/p>\n<h3>\u4e8c\u3001\u6838\u5fc3\u601d\u60f3&#xff1a;\u52a8\u6001\u89c4\u5212\u7684 &#034;\u677e\u5f1b&#034; \u9b54\u6cd5<\/h3>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Floyd \u7b97\u6cd5\u7684\u672c\u8d28\u662f\u52a8\u6001\u89c4\u5212&#xff0c;\u5b83\u7684\u6838\u5fc3\u601d\u8def\u53ef\u4ee5\u7528\u4e00\u53e5\u8bdd\u6982\u62ec&#xff1a;<br \/>\n&#034;\u4ece i \u5230 j \u7684\u6700\u77ed\u8def\u5f84&#xff0c;\u8981\u4e48\u76f4\u63a5\u8d70 i-&gt;j&#xff0c;\u8981\u4e48\u7ecf\u8fc7\u67d0\u4e2a\u4e2d\u95f4\u70b9 k \u8d70 i-&gt;k-&gt;j&#xff0c;\u53d6\u4e24\u8005\u4e2d\u66f4\u5c0f\u7684&#034;\u7528\u516c\u5f0f\u8868\u793a\u5c31\u662f&#xff1a;<\/p>\n<p>dist[i][j] &#061; min(dist[i][j], dist[i][k] &#043; dist[k][j])<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u8fd9\u91cc\u7684dist[i][j]\u8868\u793a\u4ece i \u5230 j \u7684\u5f53\u524d\u6700\u77ed\u8ddd\u79bb&#xff0c;\u6211\u4eec\u901a\u8fc7\u4e0d\u65ad\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u4e2d\u95f4\u70b9 k&#xff0c;\u6765\u66f4\u65b0\u8fd9\u4e2a\u8ddd\u79bb\u3002<\/p>\n<h3>\u4e09\u3001C&#043;&#043; \u4ee3\u7801\u5b9e\u73b0&#xff1a;\u4ece 0 \u5230 1 \u5199 Floyd<\/h3>\n<p>\u54b1\u4eec\u7528\u4e00\u4e2a\u5177\u4f53\u4f8b\u5b50\u6765\u5b9e\u73b0&#xff1a;\u5047\u8bbe\u56fe\u6709 4 \u4e2a\u8282\u70b9&#xff08;0-3&#xff09;&#xff0c;\u90bb\u63a5\u77e9\u9635\u5982\u4e0b&#xff08;INF \u8868\u793a\u4e0d\u53ef\u76f4\u8fbe&#xff09;&#xff1a;<\/p>\n<table border=\"1\" cellpadding=\"1\" cellspacing=\"1\">\n<tbody>\n<tr>\n<td>\u8d77\u70b9 \\\\ \u7ec8\u70b9<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td>5<\/td>\n<td>INF<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>INF<\/td>\n<td>0<\/td>\n<td>3<\/td>\n<td>INF<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>INF<\/td>\n<td>INF<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u6b65\u9aa4 1&#xff1a;\u521d\u59cb\u5316\u56fe&#xff08;\u90bb\u63a5\u77e9\u9635&#xff09;:<\/h4>\n<p>#include &lt;iostream&gt;<br \/>\n#include &lt;vector&gt;<br \/>\nusing namespace std;<\/p>\n<p>const int INF &#061; 1e9; \/\/ \u7528\u4e00\u4e2a\u5f88\u5927\u7684\u6570\u8868\u793a\u4e0d\u53ef\u8fbe<\/p>\n<p>int main() {<br \/>\n    \/\/ \u8282\u70b9\u6570\u91cf<br \/>\n    int n &#061; 4;<\/p>\n<p>    \/\/ \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635<br \/>\n    vector&lt;vector&lt;int&gt;&gt; dist(n, vector&lt;int&gt;(n, INF));<\/p>\n<p>    \/\/ \u81ea\u8eab\u5230\u81ea\u8eab\u7684\u8ddd\u79bb\u4e3a0<br \/>\n    for (int i &#061; 0; i &lt; n; i&#043;&#043;) {<br \/>\n        dist[i][i] &#061; 0;<br \/>\n    }<\/p>\n<p>    \/\/ \u586b\u5145\u8fb9\u7684\u6743\u91cd&#xff08;\u6839\u636e\u4e0a\u9762\u7684\u8868\u683c&#xff09;<br \/>\n    dist[0][1] &#061; 5;<br \/>\n    dist[0][3] &#061; 10;<br \/>\n    dist[1][2] &#061; 3;<br \/>\n    dist[2][3] &#061; 1;<\/p>\n<p>    \/\/ \u63a5\u4e0b\u6765\u5c31\u662fFloyd\u7684\u6838\u5fc3\u5566&#xff01;<br \/>\n}<\/p>\n<h4>\u6b65\u9aa4 2&#xff1a;Floyd \u6838\u5fc3\u4e09\u91cd\u5faa\u73af:<\/h4>\n<p>\/\/ k\u662f\u4e2d\u95f4\u70b9<br \/>\nfor (int k &#061; 0; k &lt; n; k&#043;&#043;) {<br \/>\n    \/\/ i\u662f\u8d77\u70b9<br \/>\n    for (int i &#061; 0; i &lt; n; i&#043;&#043;) {<br \/>\n        \/\/ j\u662f\u7ec8\u70b9<br \/>\n        for (int j &#061; 0; j &lt; n; j&#043;&#043;) {<br \/>\n            \/\/ \u677e\u5f1b\u64cd\u4f5c&#xff1a;\u5982\u679ci-&gt;k-&gt;j\u6bd4i-&gt;j\u66f4\u77ed&#xff0c;\u5c31\u66f4\u65b0<br \/>\n            if (dist[i][k] &#043; dist[k][j] &lt; dist[i][j]) {<br \/>\n                dist[i][j] &#061; dist[i][k] &#043; dist[k][j];<br \/>\n            }<br \/>\n        }<br \/>\n    }<br \/>\n}<\/p>\n<p>\u5212\u91cd\u70b9&#xff1a;\u5916\u5c42\u5faa\u73af\u5fc5\u987b\u662f\u4e2d\u95f4\u70b9 k&#xff01;\u56e0\u4e3a\u6211\u4eec\u9700\u8981\u5148\u786e\u5b9a\u4e2d\u95f4\u70b9&#xff0c;\u518d\u66f4\u65b0\u6240\u6709 i \u5230 j \u7684\u8def\u5f84\u3002<\/p>\n<h4>\u6b65\u9aa4 3&#xff1a;\u8f93\u51fa\u7ed3\u679c:<\/h4>\n<p>\/\/ \u6253\u5370\u6240\u6709\u8282\u70b9\u5bf9\u7684\u6700\u77ed\u8def\u5f84<br \/>\ncout &lt;&lt; &#034;\u5404\u8282\u70b9\u95f4\u7684\u6700\u77ed\u8ddd\u79bb&#xff1a;&#034; &lt;&lt; endl;<br \/>\nfor (int i &#061; 0; i &lt; n; i&#043;&#043;) {<br \/>\n    for (int j &#061; 0; j &lt; n; j&#043;&#043;) {<br \/>\n        if (dist[i][j] &#061;&#061; INF) {<br \/>\n            cout &lt;&lt; &#034;INF &#034;;<br \/>\n        } else {<br \/>\n            cout &lt;&lt; dist[i][j] &lt;&lt; &#034;   &#034;;<br \/>\n        }<br \/>\n    }<br \/>\n    cout &lt;&lt; endl;<br \/>\n}<\/p>\n<h4>\u5b8c\u6574\u4ee3\u7801\u8fd0\u884c\u7ed3\u679c<\/h4>\n<p>\u5404\u8282\u70b9\u95f4\u7684\u6700\u77ed\u8ddd\u79bb&#xff1a;<br \/>\n0   5   8   9<br \/>\nINF 0   3   4<br \/>\nINF INF 0   1<br \/>\nINF INF INF 0   <\/p>\n<p>\u6765\u9a8c\u8bc1\u4e00\u4e0b&#xff1a;\u6bd4\u5982 0 \u5230 2 \u7684\u6700\u77ed\u8def\u5f84&#xff0c;0-&gt;1-&gt;2&#xff08;5&#043;3&#061;8&#xff09;&#xff0c;\u786e\u5b9e\u6bd4\u76f4\u63a5\u8d70&#xff08;INF&#xff09;\u66f4\u77ed&#xff1b;0 \u5230 3 \u7684\u6700\u77ed\u8def\u5f84\u662f 0-&gt;1-&gt;2-&gt;3&#xff08;5&#043;3&#043;1&#061;9&#xff09;&#xff0c;\u6bd4\u76f4\u63a5\u8d70 0-&gt;3&#xff08;10&#xff09;\u66f4\u77ed&#xff0c;\u5b8c\u7f8e&#xff01;<\/p>\n<h3>\u56db\u3001\u7b97\u6cd5\u5206\u6790&#xff1a;\u65f6\u95f4\u4e0e\u7a7a\u95f4\u590d\u6742\u5ea6<\/h3>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;O (n\u00b3)&#xff0c;\u56e0\u4e3a\u6709\u4e09\u91cd\u5d4c\u5957\u5faa\u73af&#xff0c;n \u662f\u8282\u70b9\u6570\u3002\u6240\u4ee5 Floyd \u7b97\u6cd5\u9002\u5408\u8282\u70b9\u6570\u8f83\u5c11\u7684\u573a\u666f&#xff08;\u6bd4\u5982 n&lt;1000&#xff09;\u3002<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;O (n\u00b2)&#xff0c;\u4e3b\u8981\u7528\u4e8e\u5b58\u50a8\u90bb\u63a5\u77e9\u9635\u3002<\/li>\n<\/ul>\n<h3>\u4e94\u3001\u8fdb\u9636\u601d\u8003&#xff1a;Floyd \u7684\u5c0f\u7ec6\u8282<\/h3>\n<li>\n<p>\u5904\u7406\u8d1f\u6743\u8fb9&#xff1a;Floyd \u7b97\u6cd5\u53ef\u4ee5\u5904\u7406\u8d1f\u6743\u8fb9&#xff0c;\u4f46\u4e0d\u80fd\u5904\u7406\u542b\u8d1f\u6743\u56de\u8def\u7684\u56fe&#xff08;\u56e0\u4e3a\u8d1f\u6743\u56de\u8def\u4f1a\u8ba9\u8def\u5f84\u957f\u5ea6\u65e0\u9650\u51cf\u5c0f&#xff09;\u3002<\/p>\n<\/li>\n<li>\n<p>\u8bb0\u5f55\u8def\u5f84&#xff1a;\u5982\u679c\u60f3\u77e5\u9053\u6700\u77ed\u8def\u5f84\u5177\u4f53\u7ecf\u8fc7\u54ea\u4e9b\u8282\u70b9&#xff0c;\u53ef\u4ee5\u989d\u5916\u7ef4\u62a4\u4e00\u4e2apath\u77e9\u9635&#xff0c;\u8bb0\u5f55 i \u5230 j \u7684\u6700\u77ed\u8def\u5f84\u4e2d&#xff0c;j \u7684\u524d\u4e00\u4e2a\u8282\u70b9\u662f\u8c01\u3002<\/p>\n<\/li>\n<li>\n<p>\u65e0\u5411\u56fe\u5904\u7406&#xff1a;\u65e0\u5411\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u662f\u5bf9\u79f0\u7684&#xff08;dist[i][j] &#061; dist[j][i]&#xff09;&#xff0c;\u521d\u59cb\u5316\u65f6\u6ce8\u610f\u53cc\u5411\u8d4b\u503c\u5373\u53ef\u3002<\/p>\n<\/li>\n<h3>\u516d\u3001\u603b\u7ed3&#xff1a;\u4ec0\u4e48\u65f6\u5019\u7528 Floyd&#xff1f;<\/h3>\n<p>\u5f53\u4f60\u9700\u8981&#xff1a;<\/p>\n<ul>\n<li>\u8ba1\u7b97\u6240\u6709\u8282\u70b9\u5bf9\u7684\u6700\u77ed\u8def\u5f84<\/li>\n<li>\u56fe\u4e2d\u8282\u70b9\u6570\u91cf\u4e0d\u591a&#xff08;n\u00b3 \u80fd\u63a5\u53d7&#xff09;<\/li>\n<li>\u53ef\u80fd\u5b58\u5728\u8d1f\u6743\u8fb9&#xff08;\u4f46\u65e0\u8d1f\u6743\u56de\u8def&#xff09;<\/li>\n<\/ul>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Floyd \u7b97\u6cd5\u7edd\u5bf9\u662f\u4f60\u7684\u4e0d\u4e8c\u4e4b\u9009&#xff01;\u5b83\u7684\u4ee3\u7801\u7b80\u6d01\u5230\u8ba9\u4eba\u60ca\u53f9&#xff0c;\u77ed\u77ed\u51e0\u884c\u5c31\u80fd\u89e3\u51b3\u590d\u6742\u7684\u591a\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u3002<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u597d\u4e86&#xff0c;\u4eca\u5929\u7684 Floyd \u7b97\u6cd5\u5c31\u8bb2\u5230\u8fd9\u91cc&#xff5e; \u5927\u5bb6\u53ef\u4ee5\u81ea\u5df1\u6539\u6539\u56fe\u7684\u8fb9\u6743&#xff0c;\u8bd5\u8bd5\u4e0d\u540c\u7684\u6848\u4f8b\u3002\u5982\u679c\u6709\u7591\u95ee&#xff0c;\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u7559\u8a00&#xff0c;\u54b1\u4eec\u4e0b\u671f\u518d\u89c1&#xff01;\u8bb0\u5f97\u70b9\u8d5e\u5173\u6ce8\u54e6&#xff5e;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb64\u6b21\u3002\u8ba1\u7b97\u6240\u6709\u8282\u70b9\u5bf9\u7684\u6700\u77ed\u8def\u5f84\u56fe\u4e2d\u8282\u70b9\u6570\u91cf\u4e0d\u591a\uff08n\u00b3 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