{"id":1467,"date":"2021-01-29T16:07:32","date_gmt":"2021-01-29T08:07:32","guid":{"rendered":"http:\/\/47.101.202.111\/?p=1467"},"modified":"2023-05-31T23:45:13","modified_gmt":"2023-05-31T15:45:13","slug":"graph-shortest-path","status":"publish","type":"post","link":"http:\/\/139.196.114.170\/?p=1467","title":{"rendered":"\u5bfb\u627e\u6700\u77ed\u8def\u5f84 – \u56fe\u8868\u793a\u4e0e\u7ecf\u5178\u7b97\u6cd5"},"content":{"rendered":"

\u6700\u77ed\u8def\u5f84<\/h2>\n\n\n\n\n\n\n\n\n\n
\u7b97\u6cd5<\/th>\n\u601d\u60f3<\/th>\n\u6c42\u89e3\u95ee\u9898<\/th>\n\u9002\u7528\u60c5\u51b5<\/th>\n\u590d\u6742\u5ea6<\/th>\n<\/tr>\n<\/thead>\n
BFS<\/td>\n\u9010\u5c42\u9012\u8fdb<\/td>\n\u76ee\u6807\u8282\u70b9\u8ddd\u79bb\u6e90\u70b9\u7684\u6700\u77ed\u8df3\u6570<\/td>\n\u65e0\u6743\u56fe\u4e2d\u8ddd\u79bb\u6e90\u70b9\u6700\u8fd1\u70b9\u3001\u6b21\u8fd1\u70b9\u2026\u2026\u6700\u8fdc\u70b9<\/td>\nO(E+V)<\/td>\n<\/tr>\n
\u62d3\u6251\u6392\u5e8f<\/td>\n\u6309\u62d3\u6251\u987a\u5e8f\u677e\u5f1b\u6bcf\u6761\u8fb9<\/td>\n\u6e90\u70b9\u5230\u76ee\u6807\u8282\u70b9\u7684\u6700\u77ed\u8def\u5f84<\/td>\n\u65e0\u73af\u52a0\u6743\u6709\u5411\u56fe<\/td>\nO(E+V)<\/td>\n<\/tr>\n
Floyd-Warshall<\/td>\n\u52a8\u6001\u89c4\u5212<\/td>\n\u4efb\u610f\u4e24\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84<\/td>\n\u5c0f\u89c4\u6a21\u56fe\uff0c\u8f93\u5165\u4e3a\u90bb\u63a5\u77e9\u9635<\/td>\nO(V^3)<\/td>\n<\/tr>\n
\u5806\u4f18\u5316\u7684Dijkstra<\/td>\n\u6bcf\u6b21\u8d2a\u5fc3\u6269\u6563\u6e90\u70b9\u6700\u8fd1\u70b9<\/td>\n\u8ddd\u79bb\u6e90\u70b9\u6240\u6709\u70b9\u7684\u6700\u77ed\u8def\u5f84<\/td>\n\u65e0\u8d1f\u6743\u5927\u89c4\u6a21\u56fe<\/td>\nO(ElogV)<\/td>\n<\/tr>\n
\u961f\u5217\u4f18\u5316\u7684Bellman-Ford<\/td>\n\u6bcf\u6b21\u6269\u6563\u88ab\u66f4\u65b0\u70b9\u7684\u90bb\u8282\u70b9<\/td>\n\u8ddd\u79bb\u6e90\u70b9\u6240\u6709\u70b9\u7684\u6700\u77ed\u8def\u5f84<\/td>\n\u6709\u8d1f\u6743\u5927\u89c4\u6a21\u56fe<\/td>\nO(E+V)\uff0c\u6700\u574fO(VE)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u94fe\u5f0f\u524d\u5411\u661f<\/h2>\n

\u56fe\u7684\u94fe\u5f0f\u524d\u5411\u661f\u8868\u793a\u6cd5\u662f\u90bb\u63a5\u8868\u7684\u6539\u8fdb\uff0c\u4ec5\u5c06\u90bb\u63a5\u8868\u7684\u8fb9\u94fe\u63a5\u6539\u4e3a\u6570\u7ec4\u5b9e\u73b0\uff0c \u5b58\u50a8\u7a7a\u95f4\u66f4\u5c0f<\/strong>\u3002vertex\u6570\u7ec4\u7d22\u5f15\u4f5c\u4e3a\u56fe\u8282\u70b9\u7f16\u53f7\uff0cedge\u6570\u7ec4\u5b58\u8fb9\uff0c\u7d22\u5f15\u4f5c\u4e3a\u6bcf\u6761\u8fb9\u7684\u5730\u5740<\/strong>\uff0c\u5b9e\u73b0\u5982\u4e0b<\/p>\n

\nint cnt;\nint vertex[V];\n\nstruct Edge {\n    int next;\n    int to;\n    double weight;\n} edge[E];\n\nvoid addEdge(int s, int to, double weight) {\n    edge[cnt].to = to;\n    edge[cnt].weight = weight;\n    edge[cnt].next = vertex[s];\n    vertex[s] = cnt++;\n}\n<\/pre>\n
BFS<\/h2>\n
int visited[V];\n\nvoid BFS(int s) {\n    queue<int> q;\n    q.push(s);\n    while(!q.empty()) {\n        int v = q.front();\n        q.pop();\n        for(edge e: v.edges())\n            if(!visited[e.to])\n                q.push(e.to);\n    }\n}<\/code><\/pre>\n
\u961f\u5217\u4f18\u5316\u7684Bellman-Ford<\/h2>\n
\u6982\u8ff0<\/h3>\n
\u7ecf\u5178\u7684Bellman-Ford\u7b97\u6cd5\u4ee5BFS\u65b9\u5f0f\u6269\u6563\u81f3\u56fe\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u548c\u6bcf\u4e00\u6761\u8fb9\uff0c\u6bcf\u8f6e\u8bbf\u95ee\u6240\u6709\u8fb9\uff0c\u66f4\u65b0\u6240\u6709\u53ef\u4ee5\u8ba1\u7b97\u51fa\u66f4\u77ed\u8def\u5f84\u7684\u8282\u70b9<\/strong>\u3002<\/p>\n
\u7ec6\u8282<\/h3>\n
\u6bcf\u8f6e\u66f4\u65b0 \u81f3\u5c11\u6709\u4e00\u4e2a\u8282\u70b9\u66f4\u65b0\u6210\u4e3a\u5230\u8fbe\u67d0\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u5b50\u8def\u5f84<\/strong> \uff0c\u66f4\u65b0\u9876\u70b9\u6570\u91cf\u8f6e\u53ef\u4fdd\u8bc1\u83b7\u5f97\u5168\u5c40\u6700\u77ed\u8def\u5f84\u3002<\/p>\n
\u8d1f\u73af\u662f\u6307\u73af\u8def\u8fb9\u6743\u503c\u548c\u4e3a\u8d1f\u6570\u3002\u73b0\u5728\u8bf4\u660e \u56fe\u4e2d\u65e0\u6e90\u70b9\u53ef\u8fbe\u7684\u8d1f\u73af\u65f6\uff0c\u5373\u4f7f\u4e0d\u8fdb\u884cBFS\u4e2d\u7684\u8bbf\u95ee\u8282\u70b9\u6807\u8bb0\u4e5f\u4e0d\u4f1a\u9677\u5165\u65e0\u9650\u5faa\u73af<\/strong>\uff1a\u7b97\u6cd5\u6bcf\u6b21\u5bfb\u5230\u8282\u70b9\u7684\u66f4\u4f18\u503c\u90fd\u4f1a\u5c06\u8282\u70b9\u91cd\u65b0\u8fdb\u5165\u961f\u5217\u6269\u6563\u66f4\u65b0\u5468\u8fb9\u8282\u70b9\u8def\u5f84\u957f\u5ea6\uff0c\u8282\u70b9\u8fbe\u6700\u4f18\u4e4b\u540e\u6240\u6709\u5bf9\u8282\u70b9\u7684\u91cd\u590d\u8bbf\u95ee\u90fd\u662f\u66f4\u957f\u8def\u5f84\u88ab\u66f4\u65b0\u6761\u4ef6\u526a\u679d\uff0c\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\u8282\u70b9\u8fbe\u5230\u6700\u4f18\u503c\u65f6\u7b97\u6cd5\u505c\u6b62<\/strong>\u3002\u4f8b\u8bbe\u6709\u5411\u8fb9\\scriptsize u\\xrightarrow{len=3}v<\/code>\uff0c\\scriptsize v\\xrightarrow{len=-1}u<\/code>\u3002\u4eceu\u5f00\u59cb\u6269\u6563\u5230v\uff0c\u518d\u7531v\u6269\u6563\u56de\u5230u\u65f6\u4e00\u5b9a\u662fu\u2192u\u7684\u66f4\u957f\u8def\u5f84\u88ab\u66f4\u65b0\u6761\u4ef6\u526a\u679d\u3002<\/p>\n
\u4ee5\u4e0b\u8bf4\u660e\u6e90\u70b9\u53ef\u8fbe\u7684\u8d1f\u73af\u68c0\u6d4b\u3002\u7531\u4e8e\u8d1f\u73af\u53ef\u4e00\u76f4\u4f7f\u73af\u8def\u6743\u503c\u548c\u7ee7\u7eed\u9012\u51cf\uff0c\u56e0\u6b64\u4f1a\u9010\u6b65\u5360\u636e\u4f18\u52bf\u6210\u4e3a\u6700\u77ed\u8def\u5f84\uff0c\u5e76\u7ee7\u7eed\u9012\u51cf \u9677\u5165\u65e0\u9650\u5faa\u73af<\/strong>\u3002\u4e0d\u5b58\u5728\u8d1f\u73af\u7684\u7ecf\u5178\u7b97\u6cd5\u4e2d\uff0c\u9876\u70b9\u6570\u8f6e\u662f\u53ef\u4ee5\u4fdd\u8bc1\u7b97\u6cd5\u7ed3\u675f\u7684\uff0c\u7531\u6b64\u53ea\u8981\u8fd0\u884c\u8f6e\u6570\u5927\u4e8e\u603b\u9876\u70b9\u6570\u8f6e\u5219\u4e00\u5b9a\u5b58\u5728\u8d1f\u73af<\/strong>\u3002\u961f\u5217\u4f18\u5316\u7684Bellman-Ford\u8f6e\u6570\u65e0\u6cd5\u663e\u5f0f\u63a7\u5236\uff0c\u4ee5\u67d0\u9876\u70b9\u91cd\u590d\u8bbf\u95ee\u6b21\u6570\u5927\u4e8e\u603b\u9876\u70b9\u6570\u6765\u5224\u65ad\u3002<\/p>\n
\u6e90\u70b9\u4e0d\u53ef\u8fbe\u7684\u8d1f\u73af\u68c0\u6d4b\u53ef\u6784\u9020\u4e00\u4e2a\u865a\u62df\u8282\u70b9v\u6765\u4f7f\u6e90\u70b9s\u548c\u56fe\u4e2d\u6240\u6709\u5176\u4ed6\u8282\u70b9u\u5747\u8054\u901a \\scriptsize s \\rightarrow v \\rightarrow u_i<\/code>\uff0c\u6210\u4e3a\u6e90\u70b9\u53ef\u8fbe\u7684\u8d1f\u73af\u68c0\u6d4b\u95ee\u9898\u3002<\/p>\n
\u4f18\u5316\u5b9e\u73b0<\/h3>\n
\u4ee5\u4e0b\u7b97\u6cd5\u4f7f\u7528\u961f\u5217\u5bf9Bellman-Ford\u7684\u8282\u70b9\u6269\u6563\u505a\u51fa\u4f18\u5316\uff0c\u6bcf\u8f6e\u4e2d \u4ec5\u6269\u6563\u83b7\u5f97\u7d2f\u79ef\u8def\u5f84\u66f4\u77ed\u7684\u8282\u70b9<\/strong>\uff0c\u907f\u514d\u4e86\u4e0d\u5fc5\u8981\u7684\u6269\u6563\u8ba1\u7b97\u3002\u8be5\u7b97\u6cd5\u53c8\u79f0\u4e3aSPFA\u3002<\/p>\n
const double INF = DBL_MAX;\nconst int E = 100;\nconst int V = 100;\ndouble dis[V];\nint vertex[V];\nint path[V];\nint inq[V];\nint num[V];\n\nvoid init() {\n    cnt = 0;\n    memset(vertex, -1, sizeof(vertex));\n    memset(path, -1, sizeof(path));\n    memset(inq, 0, sizeof(inq));\n    memset(num, 0, sizeof(num));\n    fill(dis, dis + V, INF);\n    for (int i = 0; i < E; i++)\n        edge[i].next = -1;\n}\n\nbool spfa(int s) {\n    init();\n    queue<int> q;\n    q.push(s);\n    dis[s] = 0.0;\n\n    while (!q.empty()) {\n        int v = q.front();\n        q.pop();\n        inq[v] = 0;\n        Edge e;\n\n        for (int i = vertex[v]; ~i; i = edge[i].next) {\n            e = edge[i];\n            if (dis[v] + e.weight < dis[e.to]) {\n                dis[e.to] = dis[v] + e.weight;\n                path[e.to] = v;\n                if (!inq[e.to]) {\n                    if(num[e.to]++ > V) return true;\n                    q.push(e.to);\n                    inq[e.to] = 1;\n                }\n            }\n        }\n    }\n\n    return false;\n}<\/code><\/pre>\n
\u4f18\u5148\u961f\u5217\u4f18\u5316\u7684Dijkstra<\/h2>\n
\u6982\u8ff0<\/h3>\n
Dijkstra\u7b97\u6cd5\u4ece\u6e90\u8282\u70b9\u4ee5BFS\u65b9\u5f0f\u6269\u6563\u81f3\u56fe\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u548c\u6bcf\u4e00\u6761\u8fb9\uff0c\u6bcf\u6b21\u5747\u9009\u62e9\u8ddd\u79bb\u6e90\u70b9\u6700\u8fd1\u7684\u8282\u70b9\u6269\u6563\uff0c\u901a\u8fc7\u6700\u4f18\u5b50\u7ed3\u6784\u63a8\u5e7f\u5230\u5168\u5c40\u6700\u4f18\u3002<\/p>\n
\u7ec6\u8282<\/h3>\n
\u4ece\u8fd0\u884c\u8fc7\u7a0b\u4e0a\u6765\u7406\u89e3\uff1a\u4ec5\u8003\u8651\u6e90\u5230\u76ee\u6807\u8282\u70b9\u7684\u591a\u6761\u8def\u5f84\uff0c\u6bcf\u6761\u8def\u5f84\u5747\u4f1a\u6269\u6563\u81f3\u76ee\u6807\u8282\u70b9<\/strong>\u3002\u82e5\u6700\u77ed\u8def\u5f84\u4f18\u4e8e\u5df2\u63a2\u7d22\u7684\u975e\u6700\u4f18\u8def\u5f84\u7684\u5b50\u8def\u5f84\uff0c\u7531\u4e8e\u7b97\u6cd5 \u603b\u9009\u62e9\u6700\u77ed\u8def\u5f84\u5411\u5916\u6269\u6563<\/strong>\uff0c\u6700\u77ed\u8def\u5f84\u4e00\u5b9a\u4f18\u5148\u5230\u8fbe\u76ee\u6807\u8282\u70b9\uff0c\u4e14\u4ecd\u662f\u5f53\u524d\u6240\u6709\u5df2\u63a2\u7d22\u8def\u5f84\u4e2d\u7684\u6700\u77ed\u8def\u5f84\u3002\u82e5\u975e\u6700\u4f18\u8def\u5f84\u7684\u5b50\u8def\u5f84\u4f18\u4e8e\u6700\u77ed\u8def\u5f84\uff0c\u7b97\u6cd5\u4f1a\u9009\u62e9\u975e\u6700\u4f18\u8def\u5f84\u6269\u6563\u5148\u5230\u8fbe\u76ee\u6807\u8282\u70b9\uff0c\u800c\u6b64\u65f6\u7684\u603b\u8def\u5f84\u4e00\u5b9a\u957f\u4e8e\u6700\u77ed\u8def\u5f84\u7684\u5b50\u8def\u5f84p\uff0c\u7b97\u6cd5\u4f1a\u9009\u62e9p\u6269\u6563\u81f3\u76ee\u6807\u8282\u70b9\uff0c\u4ecd\u662f\u6240\u6709\u5df2\u63a2\u7d22\u8def\u5f84\u4e2d\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n
\u5173\u4e8e\u6269\u6563\u8fc7\u7a0b\u4e2d\u4e0d\u6807\u8bb0\u5df2\u8bbf\u95ee\u8282\u70b9\u4e0d\u4f1a\u5bfc\u81f4\u65e0\u9650\u5faa\u73af\u95ee\u9898\uff0c\u7406\u7531\u540cBellman-Ford\u3002<\/p>\n
Dijkstra\u7b97\u6cd5\u5728\u56fe\u4e2d\u6709\u8d1f\u6743\u8fb9\u65f6\u53ef\u80fd\u4f1a\u5bfc\u81f4\u7b97\u6cd5\u51fa\u9519\u65e0\u6cd5\u7ee7\u7eed\u8fdb\u884c\u3002<\/p>\n
\u76ee\u6807\u8282\u70b9\u53ef\u8fbe\u6027\u95ee\u9898\uff0c\u5373\u7b97\u6cd5\u7ed3\u675f\u540e\u5224\u65ad \\scriptsize s \\xrightarrow{len=\\infty?} u<\/code> \u95ee\u9898<\/p>\n
\u4f18\u5316\u5b9e\u73b0<\/h3>\n
\u8282\u70b9\u4ece\u7b2c\u4e00\u6b21\u8bbf\u95ee\u65f6\u8fdb\u5165\u961f\u5217\uff0c\u4e4b\u540e\u8bbf\u95ee\u5230\u8be5\u8282\u70b9\u65f6 \u8981\u4e48\u8fbe\u5230\u6700\u4f18\u5df2\u51fa\u961f\u5217<\/strong> \uff0c\u5df2\u8fdb\u884c\u4e86\u6700\u4f18\u503c\u7684\u6269\u6563\u66f4\u65b0\uff0c\u65e0\u9700\u518d\u8fdb\u884c\u66f4\u65b0\u6761\u4ef6\u5224\u65ad\uff0c\u8981\u4e48\u8fd8 \u672a\u8fbe\u5230\u6700\u4f18\u5728\u961f\u5217\u4e2d<\/strong> \u672a\u5f00\u59cb\u6269\u6563\uff0c\u4ec5\u66f4\u65b0\u8282\u70b9\u503c\uff0c\u5f85\u540e\u7eed\u8fbe\u5230\u6700\u4f18\u51fa\u961f\u65f6\u6269\u6563\u66f4\u65b0\u5468\u8fb9\u8282\u70b9\u5373\u53ef\u3002\u4ee5\u4e0b\u7b97\u6cd5\u505a\u51fa\u4e24\u5904\u4f18\u5316\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ec5\u8fdb\u5165\u961f\u5217\u4e00\u6b21\uff0c\u907f\u514d\u91cd\u590d\u6269\u6563\u548c\u8ba1\u7b97<\/p>\n
const double INF = __DBL_MAX__;\nconst int V = 15;\nconst int E = 15;\ndouble dist[V];\nint path[V];\nint done[V];\nint inq[V];\n\nstruct Cmp {\n    bool operator()(int u, int v) {\n        return dist[u] > dist[v];\n    }\n};\n\nvoid init() {\n    cnt = 0;\n    memset(path, -1, sizeof(path));\n    memset(vertex, -1, sizeof(vertex));\n    memset(inq, 0, sizeof(inq));\n    memset(done, 0, sizeof(done));\n    fill(dist, dist + V, INF);\n    for (int i = 0; i < E; i++)\n        edge[i].next = -1;\n}\n\nvoid dijkstra(int s) {\n    priority_queue<int, vector<int>, Cmp> q;\n    q.push(s);\n    dist[s] = 0.0;\n\n    while (!q.empty()) {\n        int v = q.top();\n        done[v] = 1;\n        inq[v] = 0;\n        q.pop();\n        for (int i = vertex[v]; ~i; i = edge[i].next) {\n            Edge e = edge[i];\n            double w = e.weight;\n            int t = e.to;\n            if (done[t]) continue;\n            if (dist[v] + w < dist[t]) {\n                dist[t] = dist[v] + w;\n                path[t] = v;\n                if(!inq[t]) { q.push(t); inq[t] = 1;}\n            }\n        }\n    }\n}\n\nbool hasPathTo(int v) {\n    return dist[v] < INF;\n}\n\nvoid printPath(int v) {\n    int t = path[v];\n    if (~t) printPath(t);\n    printf("%d ->", v);\n}\n\nvoid printPathTo(int v) {\n    if (hasPathTo(v)) printPath(v);\n    else printf("No Path to %d", v);\n}\n\nint main() {\n    init();\n    int v, e;\n    while (~scanf("%d%d", &v, &e)) {\n        while (e--) {\n            int s, t;\n            double w;\n            scanf("%d%d%lf", &s, &t, &w);\n            addEdge(s, t, w);\n        }\n    }\n\n    int s, t;\n    dijkstra(s);\n    printPathTo(t);\n    return 0;\n}<\/code><\/pre>\n
Floyd-Warshall<\/h2>\n
\u6982\u8ff0<\/h3>\n
\u679a\u4e3e\u6240\u6709i\u5230j\u7684\u8def\u5f84i->k->j\uff0c\u53d6\u679a\u4e3e\u6700\u4f18\u503c\u66f4\u65b0<\/p>\n
\u7ec6\u8282<\/h3>\n
\u82e5i->j\u7684\u8def\u5f84\u53ef\u8868\u793a\u4e3ad[i][j]\uff0c\u5219\u6700\u77ed\u8def\u5f84\u7684\u66f4\u65b0\u6761\u4ef6\u53ef\u8868\u793a\u4e3a\\scriptsize d[i][k] + d[k][j] \\lt d[i][j]<\/code> \u52a8\u6001\u89c4\u5212\u8f6c\u79fb\u65b9\u7a0b\u4e3a \\scriptsize dp[i][j] = min(dp[i][j], dp[i][k] + d[k][j])<\/code><\/p>\n
\u5c06 k\u7f6e\u4e8e\u6700\u5916\u5c42<\/strong> \u4fdd\u8bc1k\u59cb\u7ec8\u5c0f\u4e8e\u7b49\u4e8ei,j\uff0c\u5426\u5219\u7f6e\u4e8ei,j\u5185\u90e8\u4f18\u5148\u679a\u4e3ek\u65f6\uff0c\u5b50\u95ee\u9898dp[i][k]\u4e0edp[k][j]\u8fd8\u672a\u8ba1\u7b97\u6700\u4f18\uff0c\u7834\u574f\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u72b6\u6001\u8f6c\u79fb\u4f1a\u51fa\u9519\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u82e5k\u7f6e\u4e8e\u6700\u5185\u5c42\uff0c\u5219k\u4f1a\u5148\u5927\u4e8ei\u548cj\uff0c\u5982 \\scriptsize i=2,j=4,k=6<\/code> \u65f6\uff0c\\scriptsize dp[2][4]=dp[2][6]+dp[6][2]<\/code>\uff0c\u6b64\u65f6\u6700\u4f18\u7ed3\u679cdp[i][j]\u8fd8\u672a\u679a\u4e3e\u8ba1\u7b97 \\scriptsize dp[2][6]\u3001dp[6][2]<\/code> \u6700\u4f18\u503c\u56e0\u800c\u5bfc\u81f4\u8ba1\u7b97\u7ed3\u679c\u9519\u8bef\u3002<\/p>\n
\u5f53\u56fe\u4e2d\u5b58\u5728\u8d1f\u73af\u65f6\uff0c\u7ed5\u4e00\u5708\u540e\u56de\u6765\u8def\u5f84\u6700\u77ed\uff0cdp[i][i]\u4e00\u5b9a\u662f\u8d1f\u503c<\/p>\n
\u4f18\u5316\u5b9e\u73b0<\/h3>\n
\u65e0dp[i][k]\u8fb9\u65f6\uff0c\u53ef\u51cf\u5c11\u4e00\u8f6e\u679a\u4e3e<\/p>\n
const double INF = DBL_MAX;\nconst int V = 100;\ndouble dp[V][V];\n\nvoid init() {\n    fill(dp, dp + V * V, INF);\n    int s, t;\n    double w;\n    while(~scanf("%d%d%lf", &u, &v, &w))\n        dp[u][v] = dp[v][u] = w;\n}\n\nbool hasNegativeCircle() {\n    for(int i = 0; i < V; i++)\n        if(dp[i][i] < 0)\n            return true;\n    return false;\n}\n\nvoid floyd() {\n    init();\n    for(k = 0; k < V; k++)\n        for(int i = 0; i < V; i++)\n            if(dp[i][k] != INF)\n            for(int j = 0; j < V; j++)\n                if(dp[i][k] + dp[k][j] < dp[i][j])\n                    dp[i][j] = dp[i][k] + dp[k][j];\n    hasNegativeCircle();\n}<\/code><\/pre>\n
\u641c\u7d22\u65b9\u6cd5<\/h2>\n
\u8be6\u89c1\u641c\u7d22\u8fdb\u9636\u4e2d A* \u65b9\u6cd5\u53ca\u5176\u53d8\u79cd<\/p>\n","protected":false},"excerpt":{"rendered":"
\u6700\u77ed\u8def\u5f84 \u7b97\u6cd5 \u601d\u60f3 \u6c42\u89e3\u95ee\u9898 \u9002\u7528\u60c5\u51b5 \u590d\u6742\u5ea6 BFS \u9010\u5c42\u9012\u8fdb \u76ee\u6807\u8282\u70b9\u8ddd\u79bb\u6e90\u70b9\u7684\u6700\u77ed\u8df3\u6570 \u65e0\u6743\u56fe\u4e2d\u8ddd\u79bb\u6e90 … <\/p>\n
\u7ee7\u7eed\u9605\u8bfb\u201c\u5bfb\u627e\u6700\u77ed\u8def\u5f84 – \u56fe\u8868\u793a\u4e0e\u7ecf\u5178\u7b97\u6cd5\u201d<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[12],"tags":[41,43,42],"jetpack_featured_media_url":"","_links":{"self":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1467"}],"collection":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1467"}],"version-history":[{"count":48,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1467\/revisions"}],"predecessor-version":[{"id":2837,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1467\/revisions\/2837"}],"wp:attachment":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1467"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1467"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}