{"id":88894,"date":"2026-08-01T23:08:32","date_gmt":"2026-08-01T15:08:32","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/88894.html"},"modified":"2026-08-01T23:08:32","modified_gmt":"2026-08-01T15:08:32","slug":"%e6%9c%80%e7%9f%ad%e8%b7%af%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/88894.html","title":{"rendered":"\u6700\u77ed\u8def\u95ee\u9898"},"content":{"rendered":"<h2>\u6700\u77ed\u8def<\/h2>\n<p>\u8fd9\u7bc7\u4f1a\u8bb2\u57fa\u7840\u7684\u6700\u77ed\u8def\u95ee\u9898 \u8bb2Floyd\u7b97\u6cd5 Johnson\u7b97\u6cd5 Dijkstra\u7b97\u6cd5 Bellman-Ford \/ SPFA \u7b97\u6cd5\u6700\u540e\u6bd4\u8f83\u4e00\u4e0b<\/p>\n<h2>\u76ee\u5f55<\/h2>\n<ul>\n<li>1.\u4ec0\u4e48\u662f\u6700\u77ed\u8def<\/li>\n<li>2.Floyd\u7b97\u6cd5<\/li>\n<li>3.Bellman-Ford \/ SPFA<\/li>\n<li>4.Dijkstra\u7b97\u6cd5<\/li>\n<li>5.Johnson\u7b97\u6cd5<\/li>\n<\/ul>\n<hr \/>\n<h2>\u4e00. \u4ec0\u4e48\u662f\u6700\u77ed\u8def?<\/h2>\n<p>\u6700\u77ed\u8def\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u7ecf\u5178\u95ee\u9898: \u7ed9\u4e00\u4e2a\u5e26\u6743\u56fe \u8981\u6c42\u5bfb\u627e\u4e24\u70b9\u95f4\u6743\u503c\u4e4b\u548c\u6700\u5c0f\u7684\u8def\u5f84 \u6700\u77ed\u8def\u5206\u4e3a\u4e24\u7c7b : \u5355\u6e90\u6700\u77ed\u8def: \u6c42\u5f97\u4e00\u4e2a\u6307\u5b9a\u6e90\u70b9\u5230\u6240\u6709\u5176\u4ed6\u7ed3\u70b9\u7684\u6700\u77ed\u8def \u5168\u6e90\u6700\u77ed\u8def: \u6c42\u5f97\u56fe\u4e2d\u4efb\u610f\u4e24\u4e2a\u7ed3\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8def<\/p>\n<hr \/>\n<p>\u7531\u5b9a\u4e49 \u6211\u4eec\u6613\u5f97\u6700\u77ed\u8def\u7684\u51e0\u6761\u6027\u8d28: 1. \u6709\u6700\u77ed\u8def\u7684\u56fe \u4e00\u5b9a\u4e0d\u662f\u5e26\u8d1f\u73af\u7684\u56fe (\u65e0\u9650\u7ecf\u8fc7\u8d1f\u73af\u5de6\u811a\u8e29\u53f3\u811a\u4e0a\u5929) 2. \u5bf9\u4e8e\u4e00\u4e2a\u56fe \u6700\u77ed\u8def\u5f84\u4e0d\u4f1a\u7ecf\u8fc7\u91cd\u590d\u8282\u70b9 (\u82e5\u5b58\u5728\u91cd\u590d\u8282\u70b9 \u75311\u5f97 \u4e00\u5b9a\u591a\u7684\u662f\u4e00\u6761\u6b63\u6743\u8fb9 \u6545\u4e0d\u662f\u6700\u77ed\u8def) 3.\u6700\u77ed\u8def\u6709\u6700\u4f18\u5b50\u7ed3\u6784 \u6700\u77ed\u8def\u5f84\u4e2d\u4efb\u610f\u4e24\u70b9\u7684\u6700\u77ed\u8def\u5f84\u4e00\u5b9a\u5728\u539f\u6700\u77ed\u8def\u5f84\u4e0a (\u5982\u679c\u7ed3\u70b9 x \u548c y \u5728\u67d0\u6761\u6700\u77ed\u8def s-&gt;t \u4e0a \u90a3\u4e48 x-&gt;y \u5728\u8fd9\u6761\u8def\u5f84\u4e0a\u7684\u90e8\u5206\u5c31\u662f x \u5230 y \u7684\u6700\u77ed\u8def. \u5426\u5219, \u82e5\u5b58\u5728\u4e00\u6761\u66f4\u77ed\u7684 x-&gt;y \u8def\u5f84 \u66ff\u6362\u8fdb\u53bb\u4f1a\u5f97\u5230\u4e00\u6761\u66f4\u77ed\u7684 s-&gt;t \u8def\u5f84 \u77db\u76fe)<\/p>\n<p>\u4ece\u8fd9\u4e2a\u7b2c\u4e09\u6761\u6765\u62d3\u5c55 \u6211\u4eec\u5c31\u5f97\u51fa\u4e86\u7b2c\u4e00\u79cd\u8ba1\u7b97\u6700\u77ed\u8def\u7684\u65b9\u6cd5:<\/p>\n<h2>\u4e8c. \u6700\u77ed\u8def\u7b97\u6cd5<\/h2>\n<h3>1. Floyd\u7b97\u6cd5 (\u5168\u6e90\u6700\u77ed\u8def)<\/h3>\n<p>\u6838\u5fc3\u601d\u8def: \u52a8\u6001\u89c4\u5212 \u9010\u4e2a\u5f15\u5165\u4e24\u70b9\u95f4\u7684\u4e2d\u95f4\u7ed3\u70b9, \u770b\u901a\u8fc7\u8fd9\u4e2a\u7ed3\u70b9\u80fd\u5426\u8ba9\u67d0\u4e24\u70b9\u4e4b\u95f4\u7684\u8def\u5f84\u53d8\u77ed \u521d\u59cb\u5316<\/p>\n<p>f<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><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\u6709\u8fb9<span class=\"token operator\">:<\/span> f<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> w<span class=\"token punctuation\">;<\/span><br \/>\n\u65e0\u8fb9<span class=\"token operator\">:<\/span> f<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> INF<span class=\"token punctuation\">;<\/span><\/p>\n<p>\u5b9e\u73b0 \u5b9a\u4e49\u4e00\u4e2a\u6570\u7ec4f[x][y] : x\u5230y\u7684\u6700\u77ed\u8def\u957f\u5ea6<\/p>\n<p><span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;&#061;<\/span> n<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><span class=\"token keyword\">int<\/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;&#061;<\/span> n<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><span class=\"token keyword\">int<\/span> j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\nf<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 function\">min<\/span><span class=\"token punctuation\">(<\/span>f<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> f<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> f<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><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/\u5c1d\u8bd5\u5229\u7528k\u4f7f\u8def\u5f84\u53d8\u77ed<\/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>x\u5230y\u7684\u6700\u77ed\u8def \u8ba1\u7b97\u662f\u5426\u52a0\u5165k\u8282\u70b9 \u52a0\u5165: \u8d70k (x\u5230k\u7684\u6700\u77ed\u8def&#043;k\u5230y\u7684\u6700\u77ed\u8def) \u4e0d\u52a0\u5165: \u4e0d\u8d70k \u6cbf\u7528\u4ee5\u524d\u7684\u7ed3\u679c<\/p>\n<p>\u65f6\u95f4\u590d\u6742\u5ea6 O(n<sup>3<\/sup>) \u53ef\u7528\u4e8e\u6709\u5411\/\u65e0\u5411 \u6b63\u6743\/\u8d1f\u6743\u56fe \u4f18\u70b9 : \u7b80\u5355 \u7f3a\u70b9 : \u592a\u6162<\/p>\n<hr \/>\n<h3>2.1 Bellman-Ford \u7b97\u6cd5 (\u5355\u6e90\u6700\u77ed\u8def)<\/h3>\n<p>\u6838\u5fc3\u601d\u8def : \u677e\u5f1b \u5355\u6e90\u6700\u77ed\u8def\u6570\u7ec4 dis[x] \u8868\u793a\u4ece\u5355\u4e2a\u8282\u70b9\u5230 x \u7684\u6700\u77ed\u8def \u677e\u5f1b\u64cd\u4f5c: \u5bf9\u4e8e\u4efb\u610f\u4e00\u6761\u8fb9 u-&gt;v \u82e5 dis[u]&#043;w(u-&gt;v) \u5c0f\u4e8e dis[v] \u90a3\u4e48\u66f4\u65b0\u6211\u7684 dis[v] \u5373\u6bcf\u6b21\u66f4\u65b0\u6700\u77ed\u8def<\/p>\n<p>Bellman-Ford\u7b97\u6cd5 : \u5bf9\u6240\u6709\u8fb9\u8fdb\u884c n-1\u8f6e \u677e\u5f1b (\u6700\u957f\u7684\u6700\u77ed\u8def\u5305\u542bn-1\u6761\u8fb9 \u7ecf\u8fc7n-1\u6b21\u677e\u5f1b\u540e\u4e00\u5b9a\u80fd\u627e\u5230\u6700\u77ed\u8def) \u6839\u636e\u8fd9\u4e00\u6761\u63a8\u8bba \u6211\u4eec\u53ef\u4ee5\u5229\u7528Bellman-Ford\u627e\u8d1f\u73af : \u82e5\u6211\u4eec\u5728\u7b2c n \u8f6e\u8fd8\u5728\u677e\u5f1b \u8bc1\u660e\u8fd9\u4e2a\u56fe\u6ca1\u6709\u6700\u77ed\u8def \u8bc1\u660e\u8fd9\u4e2a\u56fe\u6709\u8d1f\u73af \u8fd9\u91ccmain\u51fd\u6570\u5b58\u8fb9 \u56e0\u4e3a\u662f\u9488\u5bf9\u8fb9\u7684\u904d\u5386<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">bellmanford<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0x3f<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> dis<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u9ed8\u8ba4\u5230\u4efb\u610f\u8282\u70b9\u8ddd\u79bb\u4e3a\u6700\u5927\u503c<\/span><br \/>\ndis<span class=\"token punctuation\">[<\/span>start<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 keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/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;&#061;<\/span> n<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\">bool<\/span> ifrelax <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> e<span class=\"token operator\">:<\/span>edges<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ edges\u662f\u8fb9\u6570\u7ec4<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> INF<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">continue<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u539f\u70b9\u5230\u4e0d\u4e86<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;<\/span>e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\ndis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;<\/span>e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">;<\/span><br \/>\nifrelax <span class=\"token operator\">&#061;<\/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><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>ifrelax<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">break<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u672a\u53d1\u751f\u677e\u5f1b \u8bc1\u660e\u5f53\u524d\u5df2\u7ecf\u662f\u6700\u77ed\u8def\u4e86<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;&#061;<\/span> n<span class=\"token punctuation\">)<\/span> iffuhuan <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u5230\u4e86\u7b2cn\u8f6e\u8fd8\u5728\u677e\u5f1b \u4e00\u5b9a\u6709\u8d1f\u73af<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token punctuation\">}<\/span><\/p>\n<p>\u65f6\u95f4\u590d\u6742\u5ea6 O(n*m) \u53ef\u7528\u4e8e\u6709\u5411\/\u65e0\u5411 \u6709\u8d1f\u6743\u56fe \u53ef\u68c0\u6d4b\u8d1f\u73af<\/p>\n<h3>2.2 Bellman_Ford\u4f18\u5316 \u2013 SPFA<\/h3>\n<p>Bellman-Ford\u5bf9\u4e8e\u6bcf\u6761\u8fb9\u90fd\u65e0\u5dee\u522b\u5224\u65ad\u4e86\u662f\u5426\u8981\u677e\u5f1b \u660e\u663e\u8fd9\u5f88\u591a\u4f59 SPFA\u7528\u961f\u5217\u7ef4\u62a4\u4e86\u54ea\u4e9b\u7ed3\u70b9\u53ef\u80fd\u5f15\u8d77\u677e\u5f1b<\/p>\n<li>\u6e90\u70b9\u5165\u961f<\/li>\n<li>\u53d6\u961f\u9996 u , \u904d\u5386 u \u7684\u6240\u6709\u51fa\u8fb9\u8fdb\u884c\u677e\u5f1b (\u4f18\u5316\u5728\u8fd9)<\/li>\n<li>\u5224\u65ad\u662f\u5426\u5165\u961f v<\/li>\n<li>\u91cd\u590d\u76f4\u5230\u961f\u5217\u4e3a\u7a7a<\/li>\n<p>\u5982\u4f55\u68c0\u6d4b\u8d1f\u73af \u8bb0\u5f55\u6bcf\u4e2a\u7ed3\u70b9\u6700\u77ed\u8def\u5f84\u7ecf\u8fc7\u7684\u8fb9\u6570 cnt[v] , \u6b63\u5e38\u60c5\u51b5\u4e0b, \u6700\u77ed\u8def\u6700\u591a\u7ecf\u8fc7 n-1 \u6761\u8fb9 \u82e5 cnt[v] &gt;&#061; n \u660e\u663e \u7ecf\u8fc7\u4e86\u8d1f\u73af<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">spfa<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0x3f<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> dis<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\ndis<span class=\"token punctuation\">[<\/span>start<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\nvis<span class=\"token punctuation\">[<\/span>start<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\">\/\/ \u8bb0\u5f55start\u5728\u4e0d\u5728\u961f\u5217\u4e2d<\/span><br \/>\nqueue<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> q<span class=\"token punctuation\">;<\/span><br \/>\nq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span>start<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>q<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><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> u <span class=\"token operator\">&#061;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">front<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\nvis<span class=\"token punctuation\">[<\/span>u<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\">\/\/ u \u4e0d\u5728\u961f\u5217\u4e2d<\/span><br \/>\n<span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> e<span class=\"token operator\">:<\/span>edges<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ edges[u] \u4e3au\u7684\u51fa\u8fb9\u6570\u7ec4<\/span><br \/>\n<span class=\"token keyword\">int<\/span> v <span class=\"token operator\">&#061;<\/span> e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">,<\/span> w <span class=\"token operator\">&#061;<\/span> e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> dis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;<\/span>w<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\ndis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;<\/span>w<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u677e\u5f1b<\/span><br \/>\ncnt<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> cnt<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u7edf\u8ba1\u7ecf\u8fc7\u8fb9\u6570<\/span><br \/>\n<span class=\"token comment\">\/\/ \u901a\u8fc7\u4e86u\u7684v\u8282\u70b9 \u8fb9\u7684\u4e2a\u6570\u4e3a\u901a\u8fc7u\u7684\u8fb9\u6570\u52a01<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>cnt<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;&#061;<\/span> n<span class=\"token punctuation\">)<\/span> iffuhuan <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u5b58\u5728\u8d1f\u73af<\/span><br \/>\n<span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>vis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token comment\">\/\/ v\u4e0d\u5728\u961f\u5217\u4e2d \u5165\u961f<\/span><br \/>\nq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span>v<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\nvis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/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><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token punctuation\">}<\/span><\/p>\n<p>\u65f6\u95f4\u590d\u6742\u5ea6 \u6700\u574fO(n*m) \u5927\u90e8\u5206\u65f6\u5019\u4f18\u4e8e\u539f\u59cbBellman-Ford \u53ef\u7528\u4e8e\u6709\u5411\/\u65e0\u5411 \u6709\u8d1f\u6743\u56fe \u53ef\u68c0\u6d4b\u8d1f\u73af<\/p>\n<hr \/>\n<h3>3. Dijkstra \u7b97\u6cd5 (\u5355\u6e90\u6700\u77ed\u8def)<\/h3>\n<p>\u6838\u5fc3\u601d\u60f3: \u8d2a\u5fc3 (\u5341\u5206\u6734\u7d20\u5bf9\u5427) \u6bcf\u6b21\u4ece\u5c1a\u672a\u786e\u5b9a\u6700\u77ed\u8def\u7684\u7ed3\u70b9\u4e2d, \u9009\u62e9\u8ddd\u79bb\u6e90\u70b9\u6700\u8fd1\u7684\u7ed3\u70b9, \u5c06\u5176\u6807\u8bb0\u4e3a&#034;\u5df2\u786e\u5b9a&#034;, \u7136\u540e\u7528\u5b83\u53bb\u677e\u5f1b\u6240\u6709\u90bb\u5c45 \u8fd9\u4e2a\u8d2a\u5fc3\u601d\u8def\u4ec5\u9002\u7528\u4e8e\u65e0\u8d1f\u6743\u8fb9\u7684\u56fe \u5728\u6b63\u6743\u8fb9\u56fe\u4e2d, \u5bf9\u4e8e\u6ca1\u786e\u5b9a\u7684\u4e00\u4e2a\u79bb\u6e90\u70b9\u6700\u8fd1\u7684\u8282\u70b9 u \u5373\u8fb9su\u6700\u5c0f \u660e\u663e\u5730 \u6ca1\u6709\u4efb\u4f55\u4e00\u4e2a\u70b9\u80fd\u6bd4 u \u8fd8\u79bb\u6e90\u70b9\u8fd1 \u9664\u975e\u5b58\u5728\u8d1f\u6743\u8fb9 \u4e3e\u4e2a\u4f8b\u5b50 <img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260801150831-6a6e0befabe85.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/> \u82e5\u5b58\u5728\u8d1f\u6743\u8fb9 \u6309\u7167\u539f\u8d2a\u5fc3\u601d\u8def \u8ddd\u79bb1\u6700\u8fd1\u7684\u8282\u70b9 u \u4e3a\u8282\u70b93 \u6700\u77ed\u8def\u4e3a1 \u4f46\u5982\u679c\u6709\u8d1f\u6743\u8fb9\u7684\u52a0\u5165 \u6700\u77ed\u8def\u660e\u663e\u662f1-&gt;2-&gt;3\u4e3a-1 \u6545\u8fd9\u4e2a\u8d2a\u5fc3\u601d\u8def\u4e0d\u9002\u7528\u4e8e\u5e26\u8d1f\u6743\u8fb9\u7684\u56fe<\/p>\n<p>\u6bcf\u6b21\u8981\u627e\u6700\u5c0f\u8fb9 \u6545\u52a0\u5165\u4f18\u5148\u961f\u5217\u4f18\u5316 \u6bcf\u6b21\u53d6\u51fa\u6700\u5c0f\u503c<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">dijkstra<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0x3f<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> dis<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ dis\u6570\u7ec4\u540cbellmanford\u7528\u6cd5<\/span><br \/>\n    dis<span class=\"token punctuation\">[<\/span>start<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    priority_queue<span class=\"token operator\">&lt;<\/span>pair<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> greater<span class=\"token operator\">&lt;<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token operator\">&gt;&gt;<\/span> q<span class=\"token punctuation\">;<\/span><br \/>\n    q<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">{<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> start<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ first\u5b58\u8ddd\u79bb second\u5b58\u8282\u70b9 \u4f7f\u4f18\u5148\u961f\u5217\u6309\u8ddd\u79bb\u6392\u5e8f<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>q<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><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> u <span class=\"token operator\">&#061;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">top<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">.<\/span>second<span class=\"token punctuation\">;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>vis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">continue<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        vis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> e<span class=\"token operator\">:<\/span>g<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/g\u662f\u90bb\u63a5\u8868<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> v <span class=\"token operator\">&#061;<\/span> g<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">,<\/span> w <span class=\"token operator\">&#061;<\/span> g<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> dis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> w<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                dis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> w<span class=\"token punctuation\">;<\/span><br \/>\n                q<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">{<\/span>dis<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> v<span class=\"token punctuation\">}<\/span><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><\/p>\n<p>\u65f6\u95f4\u590d\u6742\u5ea6 O(mlogn) \u8fb9\u6743\u4e0d\u80fd\u4e3a\u8d1f \u8fd9\u79cd\u7b97\u6cd5\u4e5f\u662f\u6700\u5e38\u7528\u7684\u5355\u6e90\u6700\u77ed\u8def\u7b97\u6cd5 \u6548\u7387\u9ad8\u4e14\u601d\u8def\u6e05\u6670<\/p>\n<hr \/>\n<h3>4. Johnson \u7b97\u6cd5 (\u5168\u6e90\u6700\u77ed\u8def)<\/h3>\n<p>\u8fd9\u4e2a\u7b97\u6cd5\u662f\u7ed3\u5408Bellman-Ford \u548c Dijkstra \u7684\u601d\u60f3\u9ad8\u6548\u6c42\u89e3\u5168\u6e90\u6700\u77ed\u8def \u60f3\u6cd5 \u76ee\u6807: \u5728\u4e00\u4e2a\u53ef\u80fd\u6709\u8d1f\u6743\u8fb9\u7684\u56fe\u4e0a, \u6c42\u4efb\u610f\u4e24\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8def(\u5168\u6e90\u6700\u77ed\u8def)<\/p>\n<li>\n<p>Floyd\u7b97\u6cd5 O(n<sup>3<\/sup>): \u80fd\u5904\u7406\u8d1f\u6743, \u4f46\u592a\u6162<\/p>\n<\/li>\n<li>\n<p>N\u6b21Dijkstra\u7b97\u6cd5 O(n*mlogn): \u5728\u7a00\u758f\u56fe\u4e0a\u6781\u5feb \u4f46Dijkstra\u5904\u7406\u4e0d\u4e86\u8d1f\u6743\u8fb9<\/p>\n<\/li>\n<p>\u6240\u4ee5 \u73b0\u5728\u7684\u95ee\u9898\u5728\u4e8e \u80fd\u4e0d\u80fd\u5bf9\u8d1f\u6743\u8fb9\u8fdb\u884c\u64cd\u4f5c \u8ba9\u5b83\u53d8\u6210\u975e\u8d1f, \u7136\u540e\u8dd1n\u904dDijkstra, \u5f97\u51fa\u5168\u6e90\u6700\u77ed\u8def \u7ed9\u6bcf\u6761\u8fb9\u90fd\u52a0\u4e0a\u4e00\u4e2a\u5f88\u5927\u7684\u6570: \u660e\u663e\u9519\u8bef \u6700\u77ed\u8def\u7b49\u4e8e\u6700\u77ed\u8def\u5f84\u7684\u6743\u503c\u52a0\u7ecf\u8fc7\u8fb9\u4e58\u4ee5\u5927\u6570<\/p>\n<hr \/>\n<p>Johnson \u7b97\u6cd5: \u7ed9\u6bcf\u4e2a\u7ed3\u70b9\u8d4b\u4e88\u4e00\u4e2a&#034;\u52bf\u80fd&#034; h[v] \u628a\u8fb9\u6743 w(u,v) \u6539\u9020\u6210 w\u2019(u,v) &#061; w(u,v) &#043; h[u] &#8211; h[v] \u628a\u6bcf\u6761\u8def\u5f84\u7d2f\u52a0 \u53d1\u73b0\u6700\u540e\u4e00\u9879\u88ab\u6d88\u4e86, \u4e8e\u662f \u4ece\u8d77\u70b9 s \u5230\u7ec8\u70b9 t \u7684\u6700\u77ed\u8def\u4e3a sum(w\u2019) &#061; sum(w) &#043; h[s] &#8211; h[t] \u6545, \u5bf9\u4e8e\u540c\u4e00\u4e2a\u8d77\u70b9 s \u548c\u540c\u4e00\u4e2a\u7ec8\u70b9 t \u65e0\u8bba\u8d70\u54ea\u6761\u8def\u5f84 \u6539\u9020\u540e\u603b\u548csum (w\u2019)\u4e0e\u6539\u9020\u524d\u603b\u548csum(w)\u7684\u5dee\u503c\u6c38\u8fdc\u662f\u56fa\u5b9a\u7684\u4e3ah[s] &#8211; h[t]<\/p>\n<p>\u63a5\u4e0b\u6765\u5c31\u662f\u600e\u4e48\u627eh[v], \u4f7f\u5f97\u6240\u6709\u65b0\u8fb9\u6743 w\u2019(u,v) \u90fd\u5927\u4e8e\u7b49\u4e8e 0 \u4e86<\/p>\n<li>\u65b0\u5efa\u4e00\u4e2a\u6e90\u70b9 S, \u4ece S \u5230\u539f\u56fe\u4e2d\u7684\u6240\u6709\u7ed3\u70b9\u8fde\u4e00\u6761\u6743\u503c\u4e3a 0 \u7684\u6709\u5411\u8fb9<\/li>\n<li>\u4ee5 S \u4e3a\u8d77\u70b9 \u8fd0\u884c\u4e00\u6b21 Bellman-Ford \u7b97\u6cd5 (\u987a\u4fbf\u68c0\u6d4b\u4e00\u4e0b\u8d1f\u73af \u5982\u679c\u6709\u8d1f\u73af \u7ec8\u6b62Johnson\u7b97\u6cd5)<\/li>\n<p>\u8dd1\u5b8c\u4e4b\u540e \u6211\u4eec\u5f97\u5230\u4e86 S \u5230\u6240\u6709\u8282\u70b9\u7684 dis[v], \u660e\u663e, dis[v]\u4e00\u5b9a &lt;&#061; 0 (\u6700\u77ed\u8def\u8981\u4e48\u76f4\u63a5\u8d70 0 \u7684\u6743\u8fb9 \u8981\u4e48\u8fd8\u6709\u8d1f\u6743\u8fb9\u8d70\u4e00\u8d70)<\/p>\n<li>\u76f4\u63a5\u4ee4\u52bf\u80fd h[v] &#061; dis[v] \u5bf9\u4e8e\u56fe\u4e2d\u7684\u4efb\u610f\u4e00\u6761\u8fb9 (u, v) \u56e0\u4e3a dis[v] \u662f S \u5230 v \u7684\u6700\u77ed\u8def \u90a3\u4e48\u5fc5\u7136\u6709 dis[v] &lt;&#061; dis[u] &#043; w(u, v) \u8fd9\u662f\u6709\u5173\u4e09\u89d2\u5f62\u7684\u4e0d\u7b49\u5f0f \u5f97\u5230\u8fd9\u4e2a\u5f0f\u5b50\u540e\u79fb\u9879: w(u, v)&#043;dis[u]-dis[v] &gt;&#061; 0 , w\u2019(u, v)\u8fd9\u4e0d\u5c31\u51fa\u4e86<\/li>\n<p>johnson\u7b97\u6cd5\u6838\u5fc3\u601d\u60f3: \u91cd\u65b0\u5b9a\u4e49\u8fb9\u7684\u6743\u503c \u7136\u540e\u8dd1n\u904ddijkstra<\/p>\n<li>\u7528bellmanford\u9884\u5904\u7406 \u540c\u65f6\u68c0\u6d4b\u8d1f\u73af<\/li>\n<li>\u7528\u5f97\u5230\u7684dis[v] \u4ee4h[v] &#061; dis[v] \u904d\u5386\u6240\u6709\u8fb9 w(u,v) \u66f4\u65b0\u4e3a w(u,v) &#043; h[u] &#8211; h[v] \u6b64\u65f6\u6240\u6709\u8fb9\u90fd\u975e\u8d1f<\/li>\n<li>\u5bf9\u539f\u56fe\u4e2d\u7684\u6bcf\u4e00\u4e2a\u7ed3\u70b9\u4f5c\u4e3a\u8d77\u70b9, \u8dd1\u4e00\u6b21 dijkstra \u5f97\u5230\u4efb\u610f\u4e24\u70b9\u5728\u65b0\u56fe\u4e2d\u7684\u6700\u77ed\u8ddd\u79bb dis_new[u][v]<\/li>\n<li>\u8fd8\u539f\u771f\u5b9e\u8ddd\u79bb dis_real[u][v] &#061; dis_new[u][v] &#043; h[v] &#8211; h[u]<\/li>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">bellman_ford<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ \u6838\u5fc3\u4ee3\u7801\u57fa\u672c\u540c\u4e0a<\/span><br \/>\n    <span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> dis<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token comment\">\/\/ \u6211\u4eec\u5b9a\u4e49\u7684\u6e90\u70b9S\u5230\u6240\u6709\u70b9\u8ddd\u79bb\u4e3a0 \u7b49\u4ef7\u4e8edis\u51680<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/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;&#061;<\/span> n<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\">bool<\/span> ifrelax <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> e <span class=\"token operator\">:<\/span> edges<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">;<\/span><br \/>\n                ifrelax <span class=\"token operator\">&#061;<\/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><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>ifrelax<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">break<\/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;&#061;<\/span> n<span class=\"token punctuation\">)<\/span> iffuhuan <span class=\"token operator\">&#061;<\/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 keyword\">void<\/span> <span class=\"token function\">dijkstra<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>dist<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0x3f<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> dist<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token function\">memset<\/span><span class=\"token punctuation\">(<\/span>vis<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span> vis<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    priority_queue<span class=\"token operator\">&lt;<\/span>pair<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> greater<span class=\"token operator\">&lt;<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token operator\">&gt;<\/span> pq<span class=\"token punctuation\">;<\/span><br \/>\n    dist<span class=\"token punctuation\">[<\/span>start<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">{<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> start<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><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> u <span class=\"token operator\">&#061;<\/span> pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">top<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">.<\/span>second<span class=\"token punctuation\">;<\/span> pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>vis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">continue<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        vis<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> e<span class=\"token operator\">:<\/span>g<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ g\u662f\u90bb\u63a5\u8868<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> v <span class=\"token operator\">&#061;<\/span> e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">,<\/span> w <span class=\"token operator\">&#061;<\/span> e<span class=\"token punctuation\">.<\/span>w<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>dist<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> dist<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> w<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                dist<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> w<span class=\"token punctuation\">;<\/span><br \/>\n                pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">{<\/span>dist<span class=\"token punctuation\">[<\/span>v<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> v<span class=\"token punctuation\">}<\/span><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 keyword\">void<\/span> <span class=\"token function\">johnson<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/\/ \u6c42\u52bf\u80fd<\/span><br \/>\n    <span class=\"token function\">bellman_ford<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>iffuhuan<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u5b58\u5728\u8d1f\u73af<\/span><br \/>\n        <span class=\"token keyword\">return<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u6539\u6743 \u5efa\u65b0\u56fe<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> e<span class=\"token operator\">:<\/span>edges<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> new_w <span class=\"token operator\">&#061;<\/span> e<span class=\"token punctuation\">.<\/span>w <span class=\"token operator\">&#043;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> dis<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ dis\u7b49\u4ef7\u4e8eh\u6570\u7ec4<\/span><br \/>\n        g<span class=\"token punctuation\">[<\/span>e<span class=\"token punctuation\">.<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">push_back<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">{<\/span>e<span class=\"token punctuation\">.<\/span>v<span class=\"token punctuation\">,<\/span> new_w<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u5bf9\u6bcf\u4e2a\u6e90\u70b9\u8dd1dijkstra, \u5e76\u8fd8\u539f\u771f\u5b9e\u8ddd\u79bb<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> s <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> s <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> s<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token function\">dijkstra<\/span><span class=\"token punctuation\">(<\/span>s<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> t <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> t <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> t<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>t<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> INF<span class=\"token punctuation\">)<\/span> <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span>s<span class=\"token operator\">-&gt;<\/span>t <span class=\"token operator\">&#061;<\/span> INF<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">else<\/span> <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span>s<span class=\"token operator\">-&gt;<\/span>t <span class=\"token operator\">&#061;<\/span> dis<span class=\"token punctuation\">[<\/span>t<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> h<span class=\"token punctuation\">[<\/span>s<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> h<span class=\"token punctuation\">[<\/span>t<span class=\"token punctuation\">]<\/span><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<h2>\u603b\u7ed3<\/h2>\n<p>\u6700\u77ed\u8def\u95ee\u9898\u662f\u56fe\u8bba\u91cc\u5f88\u5e38\u89c1\u7684\u95ee\u9898 \u8fd9\u51e0\u4e2a\u90fd\u5f88\u5e38\u7528<\/p>\n<table>\n<tr>\u7b97\u6cd5\u7c7b\u578b\u65f6\u95f4\u590d\u6742\u5ea6\u8d1f\u6743\u8fb9\u8d1f\u73af\u68c0\u6d4b<\/tr>\n<tbody>\n<tr>\n<td>Floyd<\/td>\n<td>\u5168\u6e90<\/td>\n<td>O(n<sup>3<\/sup>)<\/td>\n<td>\u652f\u6301<\/td>\n<td>\u4e0d\u652f\u6301<\/td>\n<\/tr>\n<tr>\n<td>BellmanFord<\/td>\n<td>\u5355\u6e90<\/td>\n<td>O(nm)<\/td>\n<td>\u652f\u6301<\/td>\n<td>\u652f\u6301<\/td>\n<\/tr>\n<tr>\n<td>SPFA<\/td>\n<td>\u5355\u6e90<\/td>\n<td>\u6700\u574fO(nm)<\/td>\n<td>\u652f\u6301<\/td>\n<td>\u652f\u6301<\/td>\n<\/tr>\n<tr>\n<td>Dijkstra(\u5806\u4f18\u5316)<\/td>\n<td>\u5355\u6e90<\/td>\n<td>O(mlogn)<\/td>\n<td>\u4e0d\u652f\u6301<\/td>\n<td>\u4e0d\u652f\u6301<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>\u5168\u6e90<\/td>\n<td>O(nmlogn)<\/td>\n<td>\u652f\u6301<\/td>\n<td>\u652f\u6301(\u9884\u5904\u7406\u65f6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u77ed\u8def<br 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