{"id":104,"date":"2025-03-23T01:40:33","date_gmt":"2025-03-22T17:40:33","guid":{"rendered":"http:\/\/m.qzcy2.top\/?p=104"},"modified":"2025-03-28T19:46:50","modified_gmt":"2025-03-28T11:46:50","slug":"%e5%9b%be%e8%ae%ba-%e6%9c%80%e7%9f%ad%e8%b7%af%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/m.qzcy2.top\/index.php\/2025\/03\/23\/%e5%9b%be%e8%ae%ba-%e6%9c%80%e7%9f%ad%e8%b7%af%e9%97%ae%e9%a2%98\/","title":{"rendered":"\u56fe\u8bba-\u6700\u77ed\u8def\u95ee\u9898"},"content":{"rendered":"<h1>\u58f0\u660e\uff1a<\/h1>\n<p><span style=\"color: #00ccff;\">\u672c\u6587\u4ec5\u4e3a\u4f5c\u8005\u4e3a\u4e86\u590d\u4e60\u800c\u5199\uff0c\u5e76\u975e\u4f5c\u4e3a\u6559\u7a0b\u4f7f\u7528\uff01<\/span><\/p>\n<h1>\u5e38\u89c1\u7684\u6700\u77ed\u8def\u7b97\u6cd5\uff1a<\/h1>\n<p><span style=\"color: #008000;\">\u5e38\u89c1\u7684\u6700\u77ed\u8def\u7b97\u6cd5\u4e00\u822c\u6709\uff1aBFS\u3001DFS\u3001Floyd\u3001Dijkstra\u3001Bellman-Ford\u3001SPFA\u3001Johnson\u3002<\/span><\/p>\n<h1>\u5e38\u89c1\u7684\u5b58\u56fe\u65b9\u5f0f\uff1a<\/h1>\n<p>1.\u90bb\u63a5\u8868 2.\u94fe\u5f0f\u524d\u5411\u661f(\u4e60\u60ef)<\/p>\n<h2>1..\u90bb\u63a5\u8868<\/h2>\n<p>\u4f18\u70b9\uff1a1.\u7a7a\u95f4\u590d\u6742\u5ea6\u4f4e\uff0c\u9002\u5408\u5b58\u50a8\u7a00\u758f\u56fe\uff08\u8fb9\u6570\u8f83\u5c11\u7684\u56fe\uff09\u30022.\u63d2\u5165\u548c\u5220\u9664\u8fb9\u7684\u64cd\u4f5c\u6548\u7387\u8f83\u9ad8\u3002<\/p>\n<p>\u7f3a\u70b9\uff1a1.\u5386\u904d\u6240\u6709\u8fb9\u65f6\uff0c\u9700\u8981\u904d\u5386\u6bcf\u4e2a\u9876\u70b9\u7684\u94fe\u8868\uff0c\u6548\u7387\u8f83\u4f4e\u30022.\u4e0d\u9002\u5408\u5b58\u50a8\u7a20\u5bc6\u56fe\uff08\u8fb9\u6570\u63a5\u8fd1\u9876\u70b9\u6570\u7684\u5e73\u65b9\uff09\u3002<\/p>\n<p>\u9002\u5408\u573a\u666f\uff1a\u5b58\u50a8\u8fb9\u6570\u8f83\u5c11\u7684\u56fe\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u4ea4\u901a\u7f51\u7edc\u7b49\u3002<\/p>\n<h2>2..\u94fe\u5f0f\u524d\u5411\u661f<\/h2>\n<p>\u4f18\u70b9\uff1a1.\u6548\u7387\u9ad8\uff0c\u6240\u6709\u6570\u636e\u5b58\u50a8\u5728\u6570\u7ec4\u4e2d\uff0c\u8bbf\u95ee\u901f\u5ea6\u5feb\u30022.\u9002\u5408\u5b58\u50a8\u7a00\u758f\u56fe\uff0c\u540c\u65f6\u652f\u6301\u9ad8\u6548\u7684\u8fb9\u7684\u63d2\u5165\u548c\u904d\u5386\u3002<\/p>\n<p>\u7f3a\u70b9\uff1a1.\u5b9e\u73b0\u76f8\u5bf9\u590d\u6742\uff0c\u9700\u8981\u7ef4\u62a4\u591a\u4e2a\u6570\u7ec4\u30022.\u5bf9\u4e8e\u7a20\u5bc6\u56fe\uff0c\u7a7a\u95f4\u6d6a\u8d39\u53ef\u80fd\u8f83\u5927\u3002<\/p>\n<p>\u9002\u5408\u573a\u666f\uff1a\u9002\u5408\u5b58\u50a8\u8fb9\u6570\u8f83\u591a\u4f46\u4ecd\u7136\u7a00\u758f\u7684\u56fe\uff0c\u5982\u5927\u89c4\u6a21\u7684\u7f51\u7edc\u62d3\u6251\u7ed3\u6784\u3002<\/p>\n<h2>\u603b\u7ed3\uff1a<\/h2>\n<ul>\n<li>\n<div class=\"paragraph\"><strong>\u90bb\u63a5\u8868<\/strong>\uff1a\u9002\u5408\u7a00\u758f\u56fe\uff0c\u5b9e\u73b0\u7b80\u5355\uff0c\u63d2\u5165\u548c\u5220\u9664\u8fb9\u6548\u7387\u9ad8\u3002<\/div>\n<\/li>\n<li>\n<div class=\"paragraph\"><strong>\u94fe\u5f0f\u524d\u5411\u661f<\/strong>\uff1a\u9002\u5408\u7a00\u758f\u56fe\uff0c\u7a7a\u95f4\u6548\u7387\u9ad8\uff0c\u652f\u6301\u5feb\u901f\u904d\u5386\u548c\u63d2\u5165\u8fb9\u3002<\/div>\n<\/li>\n<\/ul>\n<hr \/>\n<h1>\u6700\u77ed\u8def\u57fa\u7840\u7b97\u6cd5(\u5927\u591a\u6570\u6536\u96c6\u4e3a\u677f\u5b50)<\/h1>\n<h2>1.\u79df\u7528\u6e38\u8247(\u798f\u5efa\u590f\u4ee4\u8425) <span style=\"color: #ff9900;\">\u666e\u53ca-<\/span><\/h2>\n<p><del>\u5f00\u80c3\u5c0f\u83dc<\/del><\/p>\n<ul>\n<li><span style=\"color: #008000;\">\u7ecf\u5178\u7684\u6700\u77ed\u8def\uff0c\u4e0d\u8fc7\u534a\u77e9\u9635\u7684\u5b58\u56fe\u65b9\u5f0f\u9700\u8981\u6784\u601d\u4e00\u4e0b\uff0c\u7c7b\u4f3c\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u62bd\u8c61\u6210\u4e0a\u4e09\u89d2\u77e9\u9635\u6765\u5b58\u56fe\u3002\u90bb\u63a5\u8868<span style=\"color: #0000ff;\">\u65f6\u95f4\u590d\u6742\u5ea6O(n\u00b2)<\/span> \uff0c\u94fe\u5f0f\u524d\u5411\u661f(\u5806\u4f18\u5316dijkstra) \u65f6<span style=\"color: #0000ff;\">\u95f4\u590d\u6742\u5ea6O(nlogn)<\/span><\/span><\/li>\n<\/ul>\n<h3>\u9898\u76ee\u63cf\u8ff0<\/h3>\n<ul>\n<li>\u957f\u6c5f\u6e38\u8247\u4ff1\u4e50\u90e8\u5728\u957f\u6c5f\u4e0a\u8bbe\u7f6e\u4e86 n \u4e2a\u6e38\u8247\u51fa\u79df\u7ad9 1,2,\u22ef,n\u3002\u6e38\u5ba2\u53ef\u5728\u8fd9\u4e9b\u6e38\u8247\u51fa\u79df\u7ad9\u79df\u7528\u6e38\u8247\uff0c\u5e76\u5728\u4e0b\u6e38\u7684\u4efb\u4f55\u4e00\u4e2a\u6e38\u8247\u51fa\u79df\u7ad9\u5f52\u8fd8\u6e38\u8247\u3002\u6e38\u8247\u51fa\u79df\u7ad9 i \u5230\u6e38\u8247\u51fa\u79df\u7ad9 j \u4e4b\u95f4\u7684\u79df\u91d1\u4e3a r(i,j)\uff081\u2264i&lt;j\u2264n\uff09\u3002\u8bd5\u8bbe\u8ba1\u4e00\u4e2a\u7b97\u6cd5\uff0c\u8ba1\u7b97\u51fa\u4ece\u6e38\u8247\u51fa\u79df\u7ad9 1 \u5230\u6e38\u8247\u51fa\u79df\u7ad9 n \u6240\u9700\u7684\u6700\u5c11\u79df\u91d1\u3002<br \/>\n\u8f93\u5165\u683c\u5f0f\uff1a\u7b2c\u4e00\u884c\u4e2d\u6709\u4e00\u4e2a\u6b63\u6574\u6570 n\uff0c\u8868\u793a\u6709 n \u4e2a\u6e38\u8247\u51fa\u79df\u7ad9\u3002\u63a5\u4e0b\u6765\u7684 n\u22121 \u884c\u662f\u4e00\u4e2a\u534a\u77e9\u9635 r(i,j)\uff081\u2264i&lt;j\u2264n\uff09\u3002<br \/>\n\u8f93\u51fa\u683c\u5f0f\uff1a\u8f93\u51fa\u8ba1\u7b97\u51fa\u7684\u4ece\u6e38\u8247\u51fa\u79df\u7ad9\u00a01\u00a0\u5230\u6e38\u8247\u51fa\u79df\u7ad9\u00a0n\u00a0\u6240\u9700\u7684\u6700\u5c11\u79df\u91d1\u3002<\/li>\n<\/ul>\n<h3>\u793a\u4f8b<\/h3>\n<ul>\n<li>\u00a0\u8f93\u5165\u6837\u4f8b\uff1a 3 5 15 7 \u8f93\u51fa\u6837\u4f8b\uff1a12 \u89e3\u91ca\uff1a \u7b2c\u4e00\u884c\u7684\u6570\u5b57 3 \u8868\u793a\u6709 3 \u4e2a\u6e38\u8247\u51fa\u79df\u7ad9\u3002\u63a5\u4e0b\u6765\u7684\u4e24\u884c\u8868\u793a\u79df\u91d1\u4fe1\u606f\uff1a \u7b2c\u4e8c\u884c\u7684\u6570\u5b57 5 \u548c 15 \u8868\u793a\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 2 \u7684\u79df\u91d1\u662f 5\uff0c\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 3 \u7684\u79df\u91d1\u662f 15\u3002 \u7b2c\u4e09\u884c\u7684\u6570\u5b57 7 \u8868\u793a\u4ece\u51fa\u79df\u7ad9 2 \u5230\u51fa\u79df\u7ad9 3 \u7684\u79df\u91d1\u662f 7\u3002 \u6211\u4eec\u9700\u8981\u627e\u5230\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 3 \u7684\u6700\u5c11\u79df\u91d1\u3002\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u76f4\u63a5\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 3\uff0c\u79df\u91d1\u662f 15\u3002 \u6216\u8005\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 2\uff0c\u7136\u540e\u4ece\u51fa\u79df\u7ad9 2 \u5230\u51fa\u79df\u7ad9 3\uff0c\u603b\u79df\u91d1\u662f 5 + 7 = 12\u3002 \u56e0\u6b64\uff0c\u4ece\u51fa\u79df\u7ad9 1 \u5230\u51fa\u79df\u7ad9 3 \u7684\u6700\u5c11\u79df\u91d1\u662f 12.<\/li>\n<\/ul>\n<h3>--\u5b9e\u73b0\u65b9\u5f0f<\/h3>\n<ol>\n<li>\n<h4>\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0<\/h4>\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"cpp\">#include &lt;iostream&gt;\r\n#include &lt;vector&gt;\r\n#include &lt;algorithm&gt;\r\n#include &lt;queue&gt;\r\n#include &lt;unordered_map&gt;\r\n#include &lt;map&gt;\r\n#include &lt;cstring&gt;\r\n#include &lt;cmath&gt;\r\n#include &lt;unordered_set&gt;\r\n#include &lt;set&gt;\r\n#include &lt;utility&gt;\r\n#include &lt;climits&gt;\r\n\r\n#define int long long\r\n#define PII pair&lt;int, int&gt;\r\n#define inf 0x3f3f3f\r\n#define all(v) (v.begin() , v.end())\r\n\r\nusing namespace std;\r\nconst int N = 220;\r\n\r\nint n;\r\nint dist[N] , g[N][N];\r\nbool st[N];\r\n\r\nint dijkstra()\r\n{\r\n    memset(dist , 0x3f , sizeof dist);\r\n\r\n    dist[1] = 0;\r\n    \r\n    for (int i = 1 ; i &lt;= n ; i ++ ) \r\n    {\r\n        int t = -1;\r\n        for (int j = 1 ; j &lt;= n ; j ++ )\r\n            if (!st[j] &amp;&amp; (t == -1 || dist[t] &gt; dist[j]))\r\n                t = j;\r\n        st[t] = true;\r\n\r\n        for (int j = 1 ; j &lt;= n ; j ++ )\r\n            dist[j] = min(dist[j] , dist[t] + g[t][j]);\r\n    }\r\n\r\n    return dist[n];\r\n}\r\n\r\nvoid solve() \r\n{\r\n    cin &gt;&gt; n;\r\n\r\n    memset(g , 0x3f , sizeof g);\r\n\r\n    for (int i = 1 ; i &lt; n ; i ++ ) \r\n    {\r\n        for (int j = i + 1 ; j &lt;= n ; j ++ ) \r\n        {\r\n            cin &gt;&gt; g[i][j];\r\n        }\r\n    }\r\n\r\n    cout &lt;&lt; dijkstra() &lt;&lt; endl;\r\n\r\n}\r\n\r\nsigned main() {\r\n    ios::sync_with_stdio(false);\r\n    cin.tie(nullptr), cout.tie(nullptr);\r\n\r\n    solve();\r\n    return 0;\r\n}\r\n<\/pre>\n<\/li>\n<li>\n<h4>\u94fe\u5f0f\u524d\u5411\u661f\u5b9e\u73b0<\/h4>\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"cpp\">#include &lt;iostream&gt;\r\n#include &lt;vector&gt;\r\n#include &lt;algorithm&gt;\r\n#include &lt;queue&gt;\r\n#include &lt;unordered_map&gt;\r\n#include &lt;map&gt;\r\n#include &lt;cstring&gt;\r\n#include &lt;cmath&gt;\r\n#include &lt;unordered_set&gt;\r\n#include &lt;set&gt;\r\n#include &lt;utility&gt;\r\n#include &lt;climits&gt;\r\n\r\n\/\/ #define int long long\r\n#define PII pair&lt;int, int&gt; \r\n#define inf 0x3f\r\n#define all(v) v.begin() , v.end()\r\n\r\nusing namespace std;\r\nconst int N = 1e6  + 10;\r\n\r\nint e[N] , w[N] , ne[N] , dist[N] , h[N] , idx;\r\nint n;\r\nbool st[N];\r\n\r\nvoid add(int a , int b , int c)\r\n{\r\n    e[idx] = b , w[idx] = c , ne[idx] = h[a] , h[a] = idx ++;\r\n}\r\n\r\nvoid dijkstra()\r\n{\r\n    memset(dist , 0x3f , sizeof dist);\r\n    memset(st , false , sizeof st);\r\n    dist[1] = 0;\r\n\r\n    priority_queue&lt;PII , vector&lt;PII&gt; , greater&lt;PII&gt;&gt; heap;\r\n    heap.push({0 , 1});\r\n\r\n    while (heap.size())\r\n    {\r\n        auto t = heap.top();\r\n        heap.pop();\r\n\r\n        int distance = t.first , index = t.second;\r\n\r\n        if (st[index]) continue;\r\n        st[index] = true;\r\n\r\n        for (int i = h[index] ; i != -1 ; i = ne[i])\r\n        {\r\n            int j = e[i];\r\n            if (dist[j] &gt; distance + w[i])\r\n            {\r\n                dist[j] = distance + w[i];\r\n                heap.push({dist[j] , j});\r\n            }\r\n        }\r\n    }\r\n}\r\n\r\nvoid solve()  \r\n{   \r\n    memset(h , -1 , sizeof h);\r\n    cin &gt;&gt; n;\r\n    for (int i = 1 ; i &lt; n ; i ++ ) \r\n    {\r\n        for (int j = i + 1 ; j &lt;= n ; j ++ )\r\n        {\r\n            int distance;\r\n            cin &gt;&gt; distance;\r\n            add(i , j , distance);\r\n        }\r\n    }    \r\n\r\n    dijkstra();\r\n\r\n    cout &lt;&lt; dist[n] &lt;&lt; endl;\r\n}\r\n\r\nsigned main() \r\n{\r\n    ios::sync_with_stdio(false);\r\n    cin.tie(nullptr), cout.tie(nullptr);\r\n\r\n    solve();\r\n    return 0;\r\n}<\/pre>\n<\/li>\n<\/ol>\n<h3>dp\u89e3\u6cd5\uff1a<\/h3>\n<p><span style=\"color: #33cccc;\">\u4e0b\u6b21\u518d\u8865&gt;-..-&lt;<\/span><\/p>\n<h2>2.\u5355\u6e90\u6700\u77ed\u8def\u6a21\u677f(dijkstra\u5806\u4f18\u5316\u6a21\u677f)<span style=\"color: #ffcc00;\"> \u63d0\u9ad8-<\/span><\/h2>\n<ul>\n<li><span style=\"color: #008000;\">dijkstra\u5806\u4f18\u5316\u7248<span style=\"color: #0000ff;\">\u65f6\u95f4\u590d\u6742O(nlogn)<\/span> , \u53ef\u8fc71e5 \u4ee5\u4e0a\u7684\u6570\u636e\u3002\u540c\u6837\u53ef\u4f7f\u7528\u6734\u7d20\u7248dijkstra\u548cspfa <span style=\"color: #0000ff;\">\u65f6\u95f4\u590d\u6742\u5ea6O(n\u00b2)<\/span>\u30021e5\u6570\u636e\u4f1a\u88ab\u5361\u3002<\/span><\/li>\n<\/ul>\n<h3>\u9898\u76ee\u63cf\u8ff0<\/h3>\n<ul>\n<li>\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9\uff0cm \u6761\u6709\u5411\u8fb9\u7684\u5e26\u975e\u8d1f\u6743\u56fe\uff0c\u8bf7\u4f60\u8ba1\u7b97\u4ece s \u51fa\u53d1\uff0c\u5230\u6bcf\u4e2a\u70b9\u7684\u8ddd\u79bb\u3002\u6570\u636e\u4fdd\u8bc1\u4f60\u80fd\u4ece s \u51fa\u53d1\u5230\u4efb\u610f\u70b9\u3002<br \/>\n\u8f93\u5165\u683c\u5f0f\uff1a\u7b2c\u4e00\u884c\u4e3a\u4e09\u4e2a\u6b63\u6574\u6570 n,m,s\u3002 \u7b2c\u4e8c\u884c\u8d77 m \u884c\uff0c\u6bcf\u884c\u4e09\u4e2a\u975e\u8d1f\u6574\u6570 u i \u200b ,v i \u200b ,w i \u200b \uff0c\u8868\u793a\u4ece u i \u200b \u5230 v i \u200b \u6709\u4e00\u6761\u6743\u503c\u4e3a w i \u200b \u7684\u6709\u5411\u8fb9\u3002<br \/>\n\u8f93\u51fa\u683c\u5f0f\uff1a\u8f93\u51fa\u4e00\u884c n \u4e2a\u7a7a\u683c\u5206\u9694\u7684\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a s \u5230\u6bcf\u4e2a\u70b9\u7684\u8ddd\u79bb\u3002<\/li>\n<\/ul>\n<h3>\u793a\u4f8b<\/h3>\n<ul>\n<li>\u8f93\u5165\uff1a<br \/>\n4 6 1<br \/>\n1 2 2<br \/>\n2 3 2<br \/>\n2 4 1<br \/>\n1 3 5<br \/>\n3 4 3<br \/>\n1 4 4<\/li>\n<li>\u8f93\u51fa\uff1a<br \/>\n0 2 4 3<\/li>\n<\/ul>\n<h3>\u4ee3\u7801<\/h3>\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"cpp\">const int N = 1e6 + 10;\r\nint e[N] , ne[N] , h[N] , w[N] , dist[N] , idx;\r\n\r\nvoid add(int a , int b , int c)\r\n{\r\n    e[idx] = b , w[idx] = c , ne[idx] = h[a] , h[a] = idx ++;\r\n}\r\nbool st[N];\r\n\r\nvoid dijkstra(int s)\r\n{\r\n    memset(dist , 0x3f , sizeof dist);\r\n    memset(st , false , sizeof st);\r\n\r\n    dist[s] = 0;\r\n    priority_queue&lt;PII , vector&lt;PII&gt; , greater&lt;PII&gt; &gt; heap;\r\n    heap.push({0 , s});\r\n\r\n    while (heap.size())\r\n    {\r\n        auto t = heap.top();\r\n        heap.pop();\r\n\r\n        int distance = t.first , index = t.second;\r\n\r\n        if (st[index]) continue;\r\n        st[index] = true;\r\n\r\n        for (int i = h[index] ; ~i ; i = ne[i])\r\n        {\r\n            int j = e[i];\r\n            if (dist[j] &gt; distance + w[i])\r\n            {\r\n                dist[j] = distance + w[i];\r\n                heap.push({dist[j] , j});\r\n            }\r\n        }\r\n    }\r\n}\r\n\r\nvoid solve()\r\n{\r\n    memset(h , -1 , sizeof h);\r\n    int n , m , s;\r\n    cin &gt;&gt; n &gt;&gt; m &gt;&gt; s;\r\n\r\n    for (int i = 1 ; i &lt;= m ; i ++ )\r\n    {\r\n        int a , b , c;\r\n        cin &gt;&gt; a &gt;&gt; b &gt;&gt; c;\r\n        add(a , b , c);\r\n    }\r\n\r\n    dijkstra(s);\r\n    \r\n    for (int i = 1 ; i &lt;= n ; i ++ )\r\n        cout &lt;&lt; dist[i] &lt;&lt; \" \";\r\n}<\/pre>\n<h2>3.\u8d1f\u73af(spfa) <span style=\"color: #ffcc00;\">\u63d0\u9ad8-<\/span><\/h2>\n<ul>\n<li><span style=\"color: #008000;\">\u8bb0\u5f55\u6bcf\u6b21\u66f4\u65b0\u6700\u77ed\u8def\u52b2\u7684\u4e2a\u6570\uff0c\u5982\u679c\u66f4\u65b0\u5927\u4e8e\u7b49\u4e8en \u5219\u8bf4\u660e\u51fa\u73b0\u4e86\u8d1f\u73af\u3002\u5927\u4e8en\u5219\u8bf4\u660e\u6709\u81f3\u5c11\u4e00\u4e2a\u70b9\u4f7f\u7528\u4e86\u591a\u6b21\uff0c\u4f7f\u7528\u591a\u6b21\u53ea\u80fd\u662f\u6bcf\u6b21\u66f4\u65b0\u6700\u77ed\u8def\u5f84\u7684\u9002\u5408\u7ecf\u8fc7\u4ed6\u90fd\u80fd\u66f4\u5c0f\uff0c\u90a3\u4e48\u53ea\u80fd\u662f\u8d1f\u73af\u3002\u4f7f\u7528spfa\u4f18\u5316\u8ba1\u6570<span style=\"color: #0000ff;\">\u65f6\u95f4\u590d\u6742\u5ea6 O(n\u00b2)<\/span><\/span><\/li>\n<\/ul>\n<h3>\u9898\u76ee\u63cf\u8ff0<\/h3>\n<ul>\n<li>\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9\u7684\u6709\u5411\u56fe\uff0c\u8bf7\u6c42\u51fa\u56fe\u4e2d\u662f\u5426\u5b58\u5728\u4ece\u9876\u70b9 1 \u51fa\u53d1\u80fd\u5230\u8fbe\u7684\u8d1f\u73af\u3002 \u8d1f\u73af\u7684\u5b9a\u4e49\u662f\uff1a\u4e00\u6761\u8fb9\u6743\u4e4b\u548c\u4e3a\u8d1f\u6570\u7684\u56de\u8def<br \/>\n\u8f93\u5165\u683c\u5f0f \uff1a\u672c\u9898\u5355\u6d4b\u8bd5\u70b9\u6709\u591a\u7ec4\u6d4b\u8bd5\u6570\u636e\u3002 \u8f93\u5165\u7684\u7b2c\u4e00\u884c\u662f\u4e00\u4e2a\u6574\u6570 T\uff0c\u8868\u793a\u6d4b\u8bd5\u6570\u636e\u7684\u7ec4\u6570\u3002\u5bf9\u4e8e\u6bcf\u7ec4\u6570\u636e\u7684\u683c\u5f0f\u5982\u4e0b\uff1a<br \/>\n\u7b2c\u4e00\u884c\u6709\u4e24\u4e2a\u6574\u6570\uff0c\u5206\u522b\u8868\u793a\u56fe\u7684\u70b9\u6570 n \u548c\u63a5\u4e0b\u6765\u7ed9\u51fa\u8fb9\u4fe1\u606f\u7684\u6761\u6570 m\u3002 \u63a5\u4e0b\u6765 m \u884c\uff0c\u6bcf\u884c\u4e09\u4e2a\u6574\u6570 u,v,w\u3002 \u82e5 w\u22650\uff0c\u5219\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece u \u81f3 v \u8fb9\u6743\u4e3a w \u7684\u8fb9\uff0c\u8fd8\u5b58\u5728\u4e00\u6761\u4ece v \u81f3 u \u8fb9\u6743\u4e3a w \u7684\u8fb9\u3002 \u82e5 w&lt;0\uff0c\u5219\u53ea\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece u \u81f3 v \u8fb9\u6743\u4e3a w \u7684\u8fb9\u3002<br \/>\n\u8f93\u51fa\u683c\u5f0f\uff1a \u5bf9\u4e8e\u6bcf\u7ec4\u6570\u636e\uff0c\u8f93\u51fa\u4e00\u884c\u4e00\u4e2a\u5b57\u7b26\u4e32\uff0c\u82e5\u6240\u6c42\u8d1f\u73af\u5b58\u5728\uff0c\u5219\u8f93\u51fa YES\uff0c\u5426\u5219\u8f93\u51fa NO<\/li>\n<\/ul>\n<h3>\u793a\u4f8b<\/h3>\n<ul>\n<li>\u8f93\u5165\uff1a<br \/>\n2<br \/>\n3 4<br \/>\n1 2 2<br \/>\n1 3 4<br \/>\n2 3 1<br \/>\n3 1 -3<br \/>\n3 3<br \/>\n1 2 3<br \/>\n2 3 4<br \/>\n3 1 -8<\/li>\n<li>\u8f93\u51fa\uff1a<br \/>\nNO<br \/>\nYES<\/li>\n<\/ul>\n<h3>\u4ee3\u7801<\/h3>\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"cpp\">const int N = 1e4 + 10 , M = 4 * N;\r\nint e[M] , ne[M] , h[N] , w[M] , dist[M] , idx;\r\nbool st[M];\r\nint cnt[M];\r\nint n , m;\r\n\r\nvoid add(int a , int b , int c)\r\n{\r\n    e[idx] = b , w[idx] = c , ne[idx] = h[a] , h[a] = idx ++ ;\r\n}\r\n\r\nbool spfa()\r\n{\r\n    memset(dist , 0x3f , sizeof dist);\r\n    memset(st , false , sizeof st);\r\n    memset(cnt , 0 , sizeof cnt);\r\n\r\n    queue&lt;int&gt; q;\r\n    q.push(1);\r\n    st[1] = true;\r\n    dist[1] = 0;\r\n\r\n    while (q.size())\r\n    {\r\n        auto t = q.front();\r\n        q.pop();\r\n        st[t] = false;\r\n\r\n        for(int i = h[t] ; i != -1 ; i = ne[i])\r\n        {\r\n            int j = e[i];\r\n            if (dist[j] &gt; dist[t] + w[i])\r\n            {\r\n                dist[j] = dist[t] + w[i];\r\n                cnt[j] = cnt[t] + 1;\r\n\r\n                if (cnt[j] &gt;= n) return true;\r\n                if(!st[j])\r\n                {\r\n                    q.push(j);\r\n                    st[j] = true;\r\n                }\r\n            }\r\n        }\r\n    }\r\n    return false;\r\n}\r\n\r\nvoid solve()\r\n{\r\n    int t;\r\n    cin &gt;&gt; t;\r\n    while (t -- )\r\n    {\r\n        memset(h , -1 , sizeof h);\r\n        cin &gt;&gt; n &gt;&gt; m;\r\n        for (int i = 1 ; i &lt;= m ; i ++ )\r\n        {\r\n            int a , b , c;\r\n            cin &gt;&gt; a &gt;&gt; b &gt;&gt; c;\r\n            add(a , b , c);\r\n            if (c &gt;= 0) add(b , a , c);\r\n        }\r\n\r\n        if (spfa()) cout &lt;&lt; \"YES\" &lt;&lt; endl;\r\n        else cout &lt;&lt; \"NO\" &lt;&lt; endl;\r\n    }\r\n}<\/pre>\n<h2>4.\u6700\u77ed\u8def\u8ba1\u6570<span style=\"color: #008000;\"> \u63d0\u9ad8\/\u666e\u53ca+<\/span><\/h2>\n<p><span style=\"color: #008000;\">\u7c7bspfa\u505a\u6cd5\u540c\u65f6\u5e26\u4e00\u70b9dp\u601d\u60f3\u3002\u5f53\u524d\u8ddf\u65b0\u7684\u6700\u77ed\u8def\u5f84\u662f\u7531\u4e0a\u4e00\u4e2a\u4f4d\u7f6e\u4e3a\u7ec8\u70b9\u8f6c\u79fb\u800c\u6765\u7684\u3002\u53ea\u9700\u8981\u4f9d\u6b21\u66f4\u65b0\u5373\u53ef\u3002\u653e\u5728\u8fd9\u4e2a\u662f\u4e3a\u4e86\u7ed9\u4ee5\u540e\u6709\u627e\u6700\u77ed\u8def\u7684\u4e2a\u6570\u7559\u4e00\u4e2a\u677f\u5b50\u3002<span style=\"color: #0000ff;\">\u65f6\u95f4\u590d\u6742\u5ea6O(n + m)\u00a0<\/span><\/span><\/p>\n<h3>\u9898\u76ee\u63cf\u8ff0<\/h3>\n<ul>\n<li>\u7ed9\u51fa\u4e00\u4e2a N \u4e2a\u9876\u70b9 M \u6761\u8fb9\u7684\u65e0\u5411\u65e0\u6743\u56fe\uff0c\u9876\u70b9\u7f16\u53f7\u4e3a 1\u223cN\u3002\u95ee\u4ece\u9876\u70b9 1 \u5f00\u59cb\uff0c\u5230\u5176\u4ed6\u6bcf\u4e2a\u70b9\u7684\u6700\u77ed\u8def\u6709\u51e0\u6761\u3002<br \/>\n\u8f93\u5165\u683c\u5f0f\uff1a\u7b2c\u4e00\u884c\u5305\u542b 2 \u4e2a\u6b63\u6574\u6570 N,M\uff0c\u4e3a\u56fe\u7684\u9876\u70b9\u6570\u4e0e\u8fb9\u6570\u3002 \u63a5\u4e0b\u6765 M \u884c\uff0c\u6bcf\u884c 2 \u4e2a\u6b63\u6574\u6570 x,y\uff0c\u8868\u793a\u6709\u4e00\u6761\u8fde\u63a5\u9876\u70b9 x \u548c\u9876\u70b9 y \u7684\u8fb9\uff0c\u8bf7\u6ce8\u610f\u53ef\u80fd\u6709\u81ea\u73af\u4e0e\u91cd\u8fb9\u3002<br \/>\n\u8f93\u51fa\u683c\u5f0f\uff1a\u5171 N \u884c\uff0c\u6bcf\u884c\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u7b2c i \u884c\u8f93\u51fa\u4ece\u9876\u70b9 1 \u5230\u9876\u70b9 i \u6709\u591a\u5c11\u6761\u4e0d\u540c\u7684\u6700\u77ed\u8def\uff0c\u7531\u4e8e\u7b54\u6848\u6709\u53ef\u80fd\u4f1a\u5f88\u5927\uff0c\u4f60\u53ea\u9700\u8981\u8f93\u51fa ans mod\u00a0 100003 \u540e\u7684\u7ed3\u679c\u5373\u53ef\u3002\u5982\u679c\u65e0\u6cd5\u5230\u8fbe\u9876\u70b9 i \u5219\u8f93\u51fa 0\u3002<\/li>\n<\/ul>\n<h3>\u793a\u4f8b<\/h3>\n<ul>\n<li>\u8f93\u5165\uff1a<br \/>\n5 7<br \/>\n1 2<br \/>\n1 3<br \/>\n2 4<br \/>\n3 4<br \/>\n2 3<br \/>\n4 5<br \/>\n4 5<\/li>\n<li>\u8f93\u51fa\uff1a<br \/>\n1<br \/>\n1<br \/>\n1<br \/>\n2<br \/>\n4<\/li>\n<\/ul>\n<h3>\u4ee3\u7801<\/h3>\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"cpp\">const int mod = 1e5 + 3 , N = 1e6 + 10 , M = 4 * N;\r\nint e[M] , ne[M] , h[N] , idx;\r\nint cnt[M] , dist[M]; \r\nint n , m;\r\n\r\nvoid add(int a , int b)\r\n{\r\n    e[idx] = b , ne[idx] = h[a] , h[a] = idx ++;\r\n}\r\n\r\nvoid bfs()\r\n{\r\n    memset(dist , -1 , sizeof dist);\r\n    queue&lt;int&gt; q;\r\n\r\n    dist[1] = 0 , cnt[1] = 1;\r\n    q.push(1);\r\n\r\n    while (q.size())\r\n    {\r\n        int index = q.front();\r\n        q.pop();\r\n\r\n        for (int i = h[index] ; ~i ; i = ne[i])\r\n        {\r\n            int j = e[i];\r\n\r\n            if (dist[j] == -1)\r\n            {\r\n                dist[j] = dist[index] + 1;\r\n                cnt[j] = cnt[index];\r\n                q.push(j);\r\n            }\r\n            else if (dist[j] == dist[index] + 1)\r\n            {\r\n                cnt[j] += cnt[index];\r\n                cnt[j] %= mod;\r\n            }\r\n            \r\n        }\r\n    }\r\n\r\n}\r\n\r\nvoid solve()\r\n{\r\n    memset(h , -1 , sizeof h);\r\n    cin &gt;&gt; n &gt;&gt; m;\r\n    for (int i = 1 ; i &lt;= m ; i ++ )\r\n    {\r\n        int a , b;\r\n        cin &gt;&gt; a &gt;&gt; b;\r\n        add(a , b) , add(b , a);\r\n    }\r\n\r\n    bfs();\r\n\r\n    for (int i = 1 ; i &lt;= n ; i ++ ) cout &lt;&lt; cnt[i] &lt;&lt; endl;\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u58f0\u660e\uff1a 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&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[7],"tags":[12,20,17,16,15,21,19,18,22,13,14],"class_list":["post-104","post","type-post","status-publish","format-standard","hentry","category-algorithm","tag-algorithm","tag-bellman-ford","tag-bfs","tag-dfs","tag-dijkstra","tag-floyd","tag-johnson","tag-spfa","tag-22","tag-13","tag-14"],"_links":{"self":[{"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/posts\/104","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/comments?post=104"}],"version-history":[{"count":18,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/posts\/104\/revisions"}],"predecessor-version":[{"id":285,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/posts\/104\/revisions\/285"}],"wp:attachment":[{"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/categories?post=104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/m.qzcy2.top\/index.php\/wp-json\/wp\/v2\/tags?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}