{"id":6,"date":"2023-01-10T13:58:00","date_gmt":"2023-01-10T05:58:00","guid":{"rendered":"https:\/\/solost.cn\/?p=6"},"modified":"2024-07-23T17:40:37","modified_gmt":"2024-07-23T09:40:37","slug":"4","status":"publish","type":"post","link":"https:\/\/solost.cn\/?p=6","title":{"rendered":"\u6700\u77ed\u8def\u95ee\u9898"},"content":{"rendered":"

\u6700\u77ed\u8def\u95ee\u9898<\/h1>\n

\"652_16807d0ce7-\u6700\u77ed\u8def\"<\/p>\n

\u6734\u7d20\u7248Dijkstra(\u7a20\u5bc6\u56fe)<\/h2>\n
s\u6570\u7ec4\uff0c\u5f53\u524d\u5df2\u7ecf\u786e\u5b9a\u6700\u77ed\u8def\u5f84\u7684\u70b9\n1    dis[1] = 0, dis[i] = \u6b63\u65e0\u7a77\n2    for(int i = 0; i <= n; i ++)  \n        t <- \u4e0d\u5728s\u4e2d\u8ddd\u79bb\u6700\u8fd1\u7684\u70b9\n        s <- t \u7528t\u66f4\u65b0\u5176\u4ed6\u70b9\u7684\u8ddd\u79bb\uff08\u4ecet\u51fa\u53bb\u7684\u6240\u6709\u7684\u8fb9\u7ec4\u6210\u7684\u8def\u5f84\u80fd\u4e0d\u80fd\u66f4\u65b0\u5176          \u4ed6\u70b9\u7684\u8ddd\u79bb \n        1 -> t -> x;\n        1 ------> x;\u5982\u679c\u524d\u4e00\u79cd\u60c5\u51b5\u7684\u8ddd\u79bb\u5c0f\u4e8e\u7b2c\u4e8c\u79cd\u5c31\u53ef\u4ee5\u66f4\u65b0\n        dis[x] > dis[t] + w;<\/code><\/pre>\n
\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c\u6240\u6709\u8fb9\u6743\u5747\u4e3a\u6b63\u503c\u3002<\/p>\n
\u8bf7\u4f60\u6c42\u51fa 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\uff0c\u5982\u679c\u65e0\u6cd5\u4ece 1 \u53f7\u70b9\u8d70\u5230 n \u53f7\u70b9\uff0c\u5219\u8f93\u51fa \u22121\u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f<\/h4>\n
\u7b2c\u4e00\u884c\u5305\u542b\u6574\u6570 n \u548c m\u3002<\/p>\n
\u63a5\u4e0b\u6765 m \u884c\u6bcf\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 x, y, z\uff0c\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece\u70b9 x \u5230\u70b9 y \u7684\u6709\u5411\u8fb9\uff0c\u8fb9\u957f\u4e3a z\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f<\/h4>\n
\u8f93\u51fa\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n
\u5982\u679c\u8def\u5f84\u4e0d\u5b58\u5728\uff0c\u5219\u8f93\u51fa \u22121\u3002<\/p>\n
\u6570\u636e\u8303\u56f4<\/h4>\n
1 \u2264 n \u2264 500,
1 \u2264 m \u2264 105,
\u56fe\u4e2d\u6d89\u53ca\u8fb9\u957f\u5747\u4e0d\u8d85\u8fc710000\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h4>\n
3 3\n1 2 2\n2 3 1\n1 3 4<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h4>\n
3<\/code><\/pre>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\nconst int N = 510;\n\nint g[N][N];\nint dist[N];\nbool st[N];\nint n, m;\nint dijkstra(){\n    memset(dist, 0x3f, sizeof dist);\n    dist[1] = 0;\n  \n    for(int i = 0; i < n; i ++)\n    {\n        int t = -1;\n        for(int j = 1; j <= n; j ++)\n            if(!st[j] && (t == -1 || dist[t] > dist[j]))\n                t = j;\n      \n      \n        st[t] = true;\n      \n        for(int j = 1; j <= n; j ++){\n            dist[j] = min(dist[j], dist[t] + g[t][j]);\n        }\n    }\n  \n    if(dist[n] == 0x3f3f3f3f) return -1;\n      \n    return dist[n];\n}\n\nint main(){\n    cin >> n >> m;\n    memset(g, 0x3f, sizeof g);\n    while(m --){\n        int a, b, c;\n        cin >> a >> b >> c;\n        g[a][b] = min(g[a][b], c);\n    }\n    cout << dijkstra();\n    return 0;\n}<\/code><\/pre>\n
\u5806\u4f18\u5316\u7248Dijkstra\uff08\u7a00\u758f\u56fe\uff09<\/h2>\n
\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c\u6240\u6709\u8fb9\u6743\u5747\u4e3a\u975e\u8d1f\u503c\u3002<\/p>\n
\u8bf7\u4f60\u6c42\u51fa 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\uff0c\u5982\u679c\u65e0\u6cd5\u4ece 1 \u53f7\u70b9\u8d70\u5230 n \u53f7\u70b9\uff0c\u5219\u8f93\u51fa \u22121\u22121\u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f<\/h4>\n
\u7b2c\u4e00\u884c\u5305\u542b\u6574\u6570 n \u548c m\u3002<\/p>\n
\u63a5\u4e0b\u6765 mm \u884c\u6bcf\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 x,y,z\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece\u70b9 x \u5230\u70b9 y \u7684\u6709\u5411\u8fb9\uff0c\u8fb9\u957f\u4e3a z\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f<\/h4>\n
\u8f93\u51fa\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a 1\u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n
\u5982\u679c\u8def\u5f84\u4e0d\u5b58\u5728\uff0c\u5219\u8f93\u51fa \u22121\u3002<\/p>\n
\u6570\u636e\u8303\u56f4<\/h4>\n
1\u2264n,m\u22641.5\u00d7105
\u56fe\u4e2d\u6d89\u53ca\u8fb9\u957f\u5747\u4e0d\u5c0f\u4e8e 0\uff0c\u4e14\u4e0d\u8d85\u8fc7 10000\u3002
\u6570\u636e\u4fdd\u8bc1\uff1a\u5982\u679c\u6700\u77ed\u8def\u5b58\u5728\uff0c\u5219\u6700\u77ed\u8def\u7684\u957f\u5ea6\u4e0d\u8d85\u8fc7 109\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h4>\n
3 3\n1 2 2\n2 3 1\n1 3 4<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h4>\n
3<\/code><\/pre>\n
#include <cstring>\n#include <iostream>\n#include <algorithm>\n#include <queue>\n\nusing namespace std;\n\ntypedef pair<int, int> PII;\n\nconst int N = 1e6 + 10;\n\nint n, m;\nint h[N], w[N], e[N], ne[N], idx;\nint dist[N];\nbool st[N];\n\nvoid add(int a, int b, int c)\n{\n    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++ ;\n}\n\nint dijkstra()\n{\n    memset(dist, 0x3f, sizeof dist);\n    dist[1] = 0;\n    priority_queue<PII, vector<PII>, greater<PII>> heap;\n    heap.push({0, 1});\/\/\u524d\u4e00\u4e2a\u662f\u8ddd\u79bb\uff0c\u540e\u4e00\u4e2a\u662f\n\n    while (heap.size())\n    {\n        auto t = heap.top();\n        heap.pop();\n\n        int ver = t.second, distance = t.first;\n\n        if (st[ver]) continue;\n        st[ver] = true;\n\n        for (int i = h[ver]; i != -1; i = ne[i])\n        {\n            int j = e[i];\n            if (dist[j] > dist[ver] + w[i])\n            {\n                dist[j] = dist[ver] + w[i];\n                heap.push({dist[j], j});\n            }\n        }\n    }\n\n    if (dist[n] == 0x3f3f3f3f) return -1;\n    return dist[n];\n}\n\nint main()\n{\n    scanf("%d%d", &n, &m);\n\n    memset(h, -1, sizeof h);\n    while (m -- )\n    {\n        int a, b, c;\n        scanf("%d%d%d", &a, &b, &c);\n        add(a, b, c);\n    }\n\n    cout << dijkstra() << endl;\n\n    return 0;\n}<\/code><\/pre>\n
Bellman_Ford(\u6781\u5c11\u6570\u60c5\u51b5\uff0c\u56fe\u4e2d\u4e0d\u5b58\u5728\u8d1f\u6743\u56de\u8def(\u8d1f\u73af))<\/h2>\n
1.for(int i = 1; i <= n; i ++)\n2.\u5907\u4efddist\u9632\u6b62\u4e32\u8054\n3.for \u6240\u6709\u8fb9 a,b,w\n4.dist[b] = min(dist[b], dist[a] + w)(\u677e\u5f1b\u64cd\u4f5c)\n \u6700\u540e\u6ee1\u8db3dist[b] <= dist[a] + w(\u4e09\u89d2\u4e0d\u7b49\u5f0f)\n \n \u6b64\u7b97\u6cd5\u4e0d\u80fd\u5b58\u5728\u8d1f\u6743\u56de\u8def\n \u662f\u5426\u5b58\u5728\u8d1f\u6743\u56de\u8def\u7684\u9a8c\u8bc1\u65b9\u6cd5\n 1.\u8fed\u4ee3K\u6b21(\u4e0d\u8d85\u8fc7k\u6761\u8fb9)\u7684\u6700\u77ed\u8def\u7684\u8ddd\u79bb\n 2.\u5982\u679c\u7b2cn\u6b21\u8fed\u4ee3\u65f6\u66f4\u65b0\u4e86\u67d0\u4e9b\u8fb9(\u8bf4\u660e\u5b58\u5728\u4e00\u6761\u6700\u77ed\u8def\u5f84\u6709n\u6761\u8fb9\uff0cn+1\u4e2a\u70b9)(\u8bf4\u660e\u4e00\u5b9a\u5b58\u5728\u8d1f\u6743\u56de\u8def)\n     \u4e00\u5171\u6709n\u4e2a\u70b9\uff0c\u4f46\u662f\u51fa\u73b0\u4e86n\u4e2a\u8fb9\uff0c\u6839\u636e\u62bd\u5c49\u539f\u7406\uff0c\u8bf4\u660e\u4e00\u5b9a\u6709\u56de\u8def\u5b58\u5728<\/code><\/pre>\n
\u6709\u8fb9\u6570\u9650\u5236\u7684\u6700\u77ed\u8def<\/h2>\n
\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c \u8fb9\u6743\u53ef\u80fd\u4e3a\u8d1f\u6570<\/strong>\u3002<\/p>\n
\u8bf7\u4f60\u6c42\u51fa\u4ece 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u591a\u7ecf\u8fc7 k \u6761\u8fb9\u7684\u6700\u77ed\u8ddd\u79bb\uff0c\u5982\u679c\u65e0\u6cd5\u4ece 1 \u53f7\u70b9\u8d70\u5230 n \u53f7\u70b9\uff0c\u8f93\u51fa impossible<\/code>\u3002<\/p>\n
\u6ce8\u610f\uff1a\u56fe\u4e2d\u53ef\u80fd \u5b58\u5728\u8d1f\u6743\u56de\u8def<\/strong> \u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f<\/h4>\n
\u7b2c\u4e00\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 n,m,k\u3002<\/p>\n
\u63a5\u4e0b\u6765 m \u884c\uff0c\u6bcf\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 x,y,z\uff0c\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece\u70b9 x \u5230\u70b9 y \u7684\u6709\u5411\u8fb9\uff0c\u8fb9\u957f\u4e3a z\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f<\/h4>\n
\u8f93\u51fa\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u4ece 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u591a\u7ecf\u8fc7 k \u6761\u8fb9\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n
\u5982\u679c\u4e0d\u5b58\u5728\u6ee1\u8db3\u6761\u4ef6\u7684\u8def\u5f84\uff0c\u5219\u8f93\u51fa impossible<\/code>\u3002<\/p>\n
\u6570\u636e\u8303\u56f4<\/h4>\n
1\u2264n,k\u2264500\uff0c
1\u2264m\u226410000,
\u4efb\u610f\u8fb9\u957f\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 10000\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h4>\n
3 3 1\n1 2 1\n2 3 1\n1 3 3<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h4>\n
3<\/code><\/pre>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\nconst int N = 510, M = 10010;\nstruct Edge\n{\n    int a, b, w;\n}edges[M];\n\nint dist[N];\nint backup[N];\nint n, m, k;\n\nvoid bellman_ford(){\n    memset(dist, 0x3f, sizeof dist);\n    dist[1] = 0;\n  \n    for(int i = 0; i < k; i ++ ){\n        memcpy(backup, dist, sizeof dist);\n        for(int j = 0; j < m; j ++){\n            auto t = edges[j];\n            dist[t.b] = min(dist[t.b], backup[t.a] + t.w);\n        }\n    }\n  \n}\n\nint main(){\n    cin >> n >> m >> k;\n    for(int i = 0; i < m; i ++){\n        int a, b, w;\n        cin >> a >> b >> w;\n        edges[i] = {a, b, w};\n    }\n  \n    bellman_ford();\n  \n    if(dist[n] > 0x3f3f3f3f \/ 2) puts("impossible");\n    else cout << dist[n] << endl;\n  \n    return 0;\n}<\/code><\/pre>\n
SPFA(Shortest Path Faster Algorithm)\uff08\u5e38\u7528\uff09<\/h2>\n
\u597d\u591a\u9898\u53ef\u4ee5\u7528SPFA\u5199Dijkstra,\u5982\u679c\u88ab\u5361\u6570\u636e\uff0c\u90a3\u4e48\u5c31\u6362\u5806\u4f18\u5316\u7248\u672c\u7684Dijkstra<\/p>\n
SPFA\u6c42\u6700\u77ed\u8def<\/h3>\n
\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c \u8fb9\u6743\u53ef\u80fd\u4e3a\u8d1f\u6570<\/strong>\u3002<\/p>\n
\u8bf7\u4f60\u6c42\u51fa 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\uff0c\u5982\u679c\u65e0\u6cd5\u4ece 1 \u53f7\u70b9\u8d70\u5230 n \u53f7\u70b9\uff0c\u5219\u8f93\u51fa impossible<\/code>\u3002<\/p>\n
\u6570\u636e\u4fdd\u8bc1\u4e0d\u5b58\u5728\u8d1f\u6743\u56de\u8def\u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f<\/h4>\n
\u7b2c\u4e00\u884c\u5305\u542b\u6574\u6570 n \u548c m\u3002<\/p>\n
\u63a5\u4e0b\u6765 m \u884c\u6bcf\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 x,y,z\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece\u70b9 x \u5230\u70b9 y \u7684\u6709\u5411\u8fb9\uff0c\u8fb9\u957f\u4e3a z\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f<\/h4>\n
\u8f93\u51fa\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n
\u5982\u679c\u8def\u5f84\u4e0d\u5b58\u5728\uff0c\u5219\u8f93\u51fa impossible<\/code>\u3002<\/p>\n
\u6570\u636e\u8303\u56f4<\/h4>\n
1\u2264n,m\u2264105,
\u56fe\u4e2d\u6d89\u53ca\u8fb9\u957f\u7edd\u5bf9\u503c\u5747\u4e0d\u8d85\u8fc7 10000\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h4>\n
3 3\n1 2 5\n2 3 -3\n1 3 4<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h4>\n
2<\/code><\/pre>\n
#include <bits\/stdc++.h>\nusing namespace std;\nconst int N = 1e5 + 10;\nint idx, e[N], h[N], ne[N], w[N];\nint dist[N];\nbool st[N];\nint n, m;\nvoid add(int a, int b, int c){\n    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++;\n}\nint spfa(){\n    memset(dist, 0x3f, sizeof dist);\n    dist[1] = 0;\n  \n    queue<int>q;\n    q.push(1);\n    st[1] = true;\n  \n    while(q.size()){\n        int t = q.front();\n        q.pop();\n      \n        st[t] = false;\n      \n        for(int i = h[t]; i != -1; i = ne[i]){\n            int j = e[i];\n            if(dist[j] > dist[t] + w[i]){\n                dist[j] = dist[t] + w[i];(\u6ce8\u610f\uff0c\u662fdist[t])\n                if(!st[j]){\n                    q.push(j);\n                    st[j] = true;\n                }\n            }\n        }\n    }\n  \n    return dist[n];\n}\n\nint main(){\n    memset(h, -1, sizeof h);\n  \n    cin >> n >> m;\n  \n    while(m -- ){\n        int a, b, w;\n        cin >> a >> b >> w;\n        add(a, b, w);\n    }\n    int t = spfa();\n    if(t == 0x3f3f3f3f) puts("impossible");\n    else cout << t;\n    return 0;\n}<\/code><\/pre>\n
SPFA\u5224\u65ad\u8d1f\u73af<\/h3>\n
\u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c \u8fb9\u6743\u53ef\u80fd\u4e3a\u8d1f\u6570<\/strong>\u3002<\/p>\n
\u8bf7\u4f60\u5224\u65ad\u56fe\u4e2d\u662f\u5426\u5b58\u5728\u8d1f\u6743\u56de\u8def\u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f<\/h4>\n
\u7b2c\u4e00\u884c\u5305\u542b\u6574\u6570 n \u548c m\u3002<\/p>\n
\u63a5\u4e0b\u6765 mm \u884c\u6bcf\u884c\u5305\u542b\u4e09\u4e2a\u6574\u6570 x,y,z\uff0c\u8868\u793a\u5b58\u5728\u4e00\u6761\u4ece\u70b9 x \u5230\u70b9 y \u7684\u6709\u5411\u8fb9\uff0c\u8fb9\u957f\u4e3a z\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f<\/h4>\n
\u5982\u679c\u56fe\u4e2d\u5b58\u5728<\/strong>\u8d1f\u6743\u56de\u8def\uff0c\u5219\u8f93\u51fa Yes<\/code>\uff0c\u5426\u5219\u8f93\u51fa No<\/code>\u3002<\/p>\n
\u6570\u636e\u8303\u56f4<\/h4>\n
1\u2264n\u22642000,
1\u2264m\u226410000,
\u56fe\u4e2d\u6d89\u53ca\u8fb9\u957f\u7edd\u5bf9\u503c\u5747\u4e0d\u8d85\u8fc7 10000\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h4>\n
3 3\n1 2 -1\n2 3 4\n3 1 -4<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h4>\n
Yes<\/code><\/pre>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\nconst int N = 2010, M = 10010;\nint idx, h[N], ne[M], e[M], w[M];\nint dist[N], cnt[N];\nbool st[N];\nint n, m;\nvoid add(int a, int b, int c){\n    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++;\n}\nbool spfa(){\n    queue<int> q; \/\/\u6c42\u8d1f\u73af\u65f6\u7531\u4e8e\u53ea\u9700\u8981\u5224\u65ad\u8d1f\u73af\u662f\u5426\u5b58\u5728\uff0c\u4e0d\u9700\u8981\u6c42\u51fa\u8def\u5f84\u7684\u5177\u4f53\u957f\u5ea6\uff0c\u56e0\u6b64dist\u6570\u7ec4\u5c31\u4e0d\u9700\u8981\u521d\u59cb\u5316\u4e86\u3002\n  \n    for(int i = 1; i <= n; i ++){\n        q.push(i);\n        st[i] = true;\n    }\n  \n    while(q.size()){\n        int t = q.front();\n        q.pop();\n      \n        st[t] = false;\n      \n        for(int i = h[t]; i != -1; i = ne[i]){\n            int j = e[i];\n            if(dist[j] > dist[t] + w[i]){\n                dist[j] = dist[t] + w[i];\n                cnt[j] = cnt[t] + 1;\n              \n                if(cnt[j] >= n) return true;\n                if(!st[j]){\n                    q.push(j);\n                    st[j] = true;\n                }\n            }\n        }\n    }\n    return false;\n}\nint main(){\n    memset(h, -1, sizeof h);\n    cin >> n >> m;\n    while(m -- ){\n        int a, b, c;\n        cin >> a >> b >> c;\n        add(a, b, c);\n    }\n    if(spfa()) puts("Yes");\n    else puts("No");\n    return 0;\n}<\/code><\/pre>\n
Floyd\uff08\u591a\u6e90\u6700\u77ed\u8def\uff09<\/h2>\n
d[k, i, j] \u8868\u793a\u4ecei\u5230j\u7ecf\u8fc7k\u4e2a\u70b9\u7684\u6700\u77ed\u8ddd\u79bb<\/p>\n
d[k, i, j] = d[k - 1, i, k] + d[k - 1, k, j];<\/code><\/pre>\n
\u53ef\u4ee5\u7528\u4e00\u91cd\u5faa\u73af\u6765\u5c06\u4e09\u7ef4\u964d\u4f4e\u7ef4\u5ea6\uff0c\u964d\u4f4e\u81f3\u4e8c\u7ef4\uff0c\u5982\u4e0b\uff1a<\/p>\n
for(int k = 1; k <= n; k ++)\n   for(int i = 1; i <= n; i ++)\n       for(int j = 1; j <= n; j ++)\n           d[i][j] = min(d[i][j], d[i][k] + d[k][j]);<\/code><\/pre>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\nconst int N = 210, INF = 1e9;\nint n, m, Q;\nint d[N][N];\nvoid floyd(){\n    for(int k = 1; k <= n; k ++)\n        for(int i = 1; i <= n; i ++)\n            for(int j = 1; j <= n; j ++)\n                d[i][j] = min(d[i][j], d[i][k] + d[k][j]);\n}\nint main(){\n    cin >> n >> m >> Q;\n    for(int i = 1; i <= n; i ++)\n        for(int j = 1; j <= n; j ++)\n            if(i == j) d[i][j] = 0;\n            else d[i][j] = INF;\n  \n    while(m -- ){\n        int a, b, w;\n        cin >> a >> b >> w;\n        d[a][b] = min(d[a][b], w);\n    }\n  \n    floyd();\n  \n    while(Q --){\n        int a, b;\n        cin >> a >> b;\n        if(d[a][b] > INF \/ 2) puts("impossible");\n        else cout << d[a][b] << endl;\n    }\n    return 0;\n}<\/code><\/pre>\n
\u5927\u4f17\u60c5\u4eba<\/h2>\n
\u4eba\u4e0e\u4eba\u4e4b\u95f4\u603b\u6709\u4e00\u70b9\u8ddd\u79bb\u611f\u3002\u6211\u4eec\u5047\u5b9a\u4e24\u4e2a\u4eba\u4e4b\u95f4\u7684\u4eb2\u5bc6\u7a0b\u5ea6\u8ddf\u4ed6\u4eec\u4e4b\u95f4\u7684\u8ddd\u79bb\u611f\u6210\u53cd\u6bd4\uff0c\u5e76\u4e14\u8ddd\u79bb\u611f\u662f\u5355\u5411\u7684\u3002\u4f8b\u5982\u5c0f\u84dd\u5bf9\u5c0f\u7ea2\u60a3\u4e86\u5355\u76f8\u601d\uff0c\u4ece\u5c0f\u84dd\u7684\u773c\u4e2d\u770b\u53bb\uff0c\u4ed6\u548c\u5c0f\u7ea2\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a 1\uff0c\u53ea\u5dee\u4e00\u5c42\u7a97\u6237\u7eb8\uff1b\u4f46\u5728\u5c0f\u7ea2\u7684\u773c\u91cc\uff0c\u5979\u548c\u5c0f\u84dd\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a 108000\uff0c\u5dee\u4e86\u5341\u4e07\u516b\u5343\u91cc\u2026\u2026 \u53e6\u5916\uff0c\u6211\u4eec\u8fdb\u4e00\u6b65\u5047\u5b9a\uff0c\u8ddd\u79bb\u611f\u5728\u8ba4\u8bc6\u7684\u4eba\u4e4b\u95f4\u662f\u53ef\u4f20\u9012\u7684\u3002\u4f8b\u5982\u5c0f\u7eff\u89c9\u5f97\u81ea\u5df1\u8ddf\u5c0f\u84dd\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a 2\uff0c\u5219\u5373\u4f7f\u5c0f\u7eff\u5e76\u4e0d\u76f4\u63a5\u8ba4\u8bc6\u5c0f\u7ea2\uff0c\u6211\u4eec\u4e5f\u9ed8\u8ba4\u5c0f\u7eff\u65e9\u665a\u4f1a\u8ba4\u8bc6\u5c0f\u7ea2\uff0c\u5e76\u4e14\u56e0\u4e3a\u8ddf\u5c0f\u84dd\u5f88\u4eb2\u8fd1\u7684\u5173\u7cfb\uff0c\u5c0f\u7eff\u4f1a\u89c9\u5f97\u81ea\u5df1\u8ddf\u5c0f\u7ea2\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a 1+2=3\u3002\u5f53\u7136\u8fd9\u5e26\u6765\u4e00\u4e2a\u95ee\u9898\uff0c\u5982\u679c\u5c0f\u7eff\u672c\u6765\u4e5f\u8ba4\u8bc6\u5c0f\u7ea2\uff0c\u6216\u8005\u4ed6\u901a\u8fc7\u5176\u4ed6\u4eba\u4e5f\u80fd\u8ba4\u8bc6\u5c0f\u7ea2\uff0c\u4f46\u901a\u8fc7\u4e0d\u540c\u6e20\u9053\u63a8\u5bfc\u51fa\u6765\u7684\u8ddd\u79bb\u611f\u4e0d\u4e00\u6837\uff0c\u8be5\u600e\u4e48\u7b97\u5462\uff1f\u6211\u4eec\u5728\u8fd9\u91cc\u505a\u4e2a\u7b80\u5355\u5b9a\u4e49\uff0c\u5c31\u5c06\u5c0f\u7eff\u5bf9\u5c0f\u7ea2\u7684\u8ddd\u79bb\u611f\u5b9a\u4e49\u4e3a\u6240\u6709\u63a8\u5bfc\u51fa\u6765\u7684\u8ddd\u79bb\u611f\u7684\u6700\u5c0f\u503c\u3002<\/p>\n
\u4e00\u4e2a\u4eba\u7684\u5f02\u6027\u7f18\u4e0d\u662f\u7531\u6700\u559c\u6b22\u4ed6\/\u5979\u7684\u90a3\u4e2a\u5f02\u6027\u51b3\u5b9a\u7684\uff0c\u800c\u662f\u7531\u5bf9\u4ed6\/\u5979\u6700\u65e0\u611f\u7684\u90a3\u4e2a\u5f02\u6027\u51b3\u5b9a\u7684\u3002\u6211\u4eec\u8bb0\u4e00\u4e2a\u4eba i<\/em> \u5728\u4e00\u4e2a\u5f02\u6027 j<\/em> \u773c\u4e2d\u7684\u8ddd\u79bb\u611f\u4e3a D~ij~<\/em>\uff1b\u5c06 i<\/em> \u7684\u201c\u5f02\u6027\u7f18\u201d\u5b9a\u4e49\u4e3a 1\/max~j\u2208S(i)~{D~ij~<\/em>}\uff0c\u5176\u4e2d S<\/em>(i<\/em>) \u662f\u76f8\u5bf9\u4e8e i<\/em> \u7684\u6240\u6709\u5f02\u6027\u7684\u96c6\u5408\u3002\u90a3\u4e48\u201c\u5927\u4f17\u60c5\u4eba\u201d\u5c31\u662f\u5f02\u6027\u7f18\u6700\u597d\uff08\u503c\u6700\u5927\uff09\u7684\u90a3\u4e2a\u4eba\u3002<\/p>\n
\u672c\u9898\u5c31\u8bf7\u4f60\u4ece\u7ed9\u5b9a\u7684\u4e00\u6279\u4eba\u4e0e\u4eba\u4e4b\u95f4\u7684\u8ddd\u79bb\u611f\u4e2d\u5206\u522b\u627e\u51fa\u4e24\u4e2a\u6027\u522b\u4e2d\u7684\u201c\u5927\u4f17\u60c5\u4eba\u201d\u3002<\/p>\n
\u8f93\u5165\u683c\u5f0f\uff1a<\/h3>\n
\u8f93\u5165\u5728\u7b2c\u4e00\u884c\u4e2d\u7ed9\u51fa\u4e00\u4e2a\u6b63\u6574\u6570 N<\/em>\uff08\u2264500\uff09\uff0c\u4e3a\u603b\u4eba\u6570\u3002\u4e8e\u662f\u6211\u4eec\u9ed8\u8ba4\u6240\u6709\u4eba\u4ece 1 \u5230 N<\/em> \u7f16\u53f7\u3002<\/p>\n
\u968f\u540e N<\/em> \u884c\uff0c\u7b2c i<\/em> \u884c\u63cf\u8ff0\u4e86\u7f16\u53f7\u4e3a i<\/em> \u7684\u4eba\u4e0e\u5176\u4ed6\u4eba\u7684\u5173\u7cfb\uff0c\u683c\u5f0f\u4e3a\uff1a<\/p>\n
\u6027\u522b K \u670b\u53cb1:\u8ddd\u79bb1 \u670b\u53cb2:\u8ddd\u79bb2 \u2026\u2026 \u670b\u53cbK:\u8ddd\u79bbK<\/code><\/pre>\n
\u5176\u4e2d \u6027\u522b<\/code> \u662f\u8fd9\u4e2a\u4eba\u7684\u6027\u522b\uff0cF<\/code> \u8868\u793a\u5973\u6027\uff0cM<\/code> \u8868\u793a\u7537\u6027\uff1bK<\/code>\uff08<N<\/em> \u7684\u975e\u8d1f\u6574\u6570\uff09\u4e3a\u8fd9\u4e2a\u4eba\u76f4\u63a5\u8ba4\u8bc6\u7684\u670b\u53cb\u6570\uff1b\u968f\u540e\u7ed9\u51fa\u7684\u662f\u8fd9 K<\/code> \u4e2a\u670b\u53cb\u7684\u7f16\u53f7\u3001\u4ee5\u53ca\u8fd9\u4e2a\u4eba\u5bf9\u8be5\u670b\u53cb\u7684\u8ddd\u79bb\u611f\u3002\u8ddd\u79bb\u611f\u662f\u4e0d\u8d85\u8fc7 106 \u7684\u6b63\u6574\u6570\u3002<\/p>\n
\u9898\u76ee\u4fdd\u8bc1\u7ed9\u51fa\u7684\u5173\u7cfb\u4e2d\u4e00\u5b9a\u4e24\u79cd\u6027\u522b\u7684\u4eba\u90fd\u6709\uff0c\u4e0d\u4f1a\u51fa\u73b0\u91cd\u590d\u7ed9\u51fa\u7684\u5173\u7cfb\uff0c\u5e76\u4e14\u6bcf\u4e2a\u4eba\u7684\u670b\u53cb\u4e2d\u90fd\u4e0d\u5305\u542b\u81ea\u5df1\u3002<\/p>\n
\u8f93\u51fa\u683c\u5f0f\uff1a<\/h3>\n
\u7b2c\u4e00\u884c\u7ed9\u51fa\u81ea\u8eab\u4e3a\u5973\u6027\u7684\u201c\u5927\u4f17\u60c5\u4eba\u201d\u7684\u7f16\u53f7\uff0c\u7b2c\u4e8c\u884c\u7ed9\u51fa\u81ea\u8eab\u4e3a\u7537\u6027\u7684\u201c\u5927\u4f17\u60c5\u4eba\u201d\u7684\u7f16\u53f7\u3002\u5982\u679c\u5b58\u5728\u5e76\u5217\uff0c\u5219\u6309\u7f16\u53f7\u9012\u589e\u7684\u987a\u5e8f\u8f93\u51fa\u6240\u6709\u3002\u6570\u5b57\u95f4\u4ee5\u4e00\u4e2a\u7a7a\u683c\u5206\u9694\uff0c\u884c\u9996\u5c3e\u4e0d\u5f97\u6709\u591a\u4f59\u7a7a\u683c\u3002<\/p>\n
\u8f93\u5165\u6837\u4f8b\uff1a<\/h3>\n
6\nF 1 4:1\nF 2 1:3 4:10\nF 2 4:2 2:2\nM 2 5:1 3:2\nM 2 2:2 6:2\nM 2 3:1 2:5<\/code><\/pre>\n
\u8f93\u51fa\u6837\u4f8b\uff1a<\/h3>\n
2 3\n4<\/code><\/pre>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\n\nconst int INF = 1e9, N = 510;\nint n;\nint sex[N], dist[N], g[N][N];\nint main(){\n    cin >> n;\n\n    for(int i = 1; i <= n; i ++)\n        for(int j = 1; j <= n; j ++)\n            if(i == j) g[i][j] = 0;\n            else g[i][j] = INF;\n        \n    for(int i = 1; i <= n; i ++){\n        char op[2];\n        int k;\n        cin >> op >> k;\n        if(*op == 'M') sex[i] = 1;\n        for(int j = 1; j <= k; j ++){\n            int a, b;\n            scanf("%d:%d", &a, &b);\n            g[i][a] = b;\n        }\n    }\n\n    for(int k = 1; k <= n; k ++)\n        for(int i = 1; i <= n; i ++)\n            for(int j = 1; j <= n; j ++)\n                g[i][j] = min(g[i][j], g[i][k] + g[k][j]);\n            \n    for(int i = 1; i <= n; i ++){\n        dist[i] = 0;\n        for(int j = 1; j <= n; j ++){\n            if(sex[i] != sex[j]){\n                dist[i] = max(dist[i], g[j][i]);\n            }\n        }\n    }\n\n    for(int i = 0; i < 2; i ++){\n        int minn = 2e9;\n        for(int j = 1; j <= n; j ++){\n            if(sex[j] == i){\n                minn = min(minn, dist[j]);\n            }\n        }\n        int f = 0;\n        for(int j = 1; j <= n; j ++){\n            if(sex[j] == i && minn == dist[j]){\n                if(f) cout << " ";\n                cout << j;\n                f ++;\n            }\n        }\n        cout << endl;\n    }\n    return 0;\n}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
\u6700\u77ed\u8def\u95ee\u9898 \u6734\u7d20\u7248Dijkstra(\u7a20\u5bc6\u56fe) s\u6570\u7ec4\uff0c\u5f53\u524d\u5df2\u7ecf\u786e\u5b9a\u6700\u77ed\u8def\u5f84\u7684\u70b9 1 dis[1] = 0, dis[i] = \u6b63\u65e0\u7a77 2 for(int i = 0; i <= n; i ++) t <- \u4e0d\u5728s\u4e2d\u8ddd\u79bb\u6700\u8fd1\u7684\u70b9 s <- t \u7528t\u66f4\u65b0\u5176\u4ed6\u70b9\u7684\u8ddd\u79bb\uff08\u4ecet\u51fa\u53bb\u7684\u6240\u6709\u7684\u8fb9\u7ec4\u6210\u7684\u8def\u5f84\u80fd\u4e0d\u80fd\u66f4\u65b0\u5176 \u4ed6\u70b9\u7684\u8ddd\u79bb 1 -> t -> x; 1 ------> x;\u5982\u679c\u524d\u4e00\u79cd\u60c5\u51b5\u7684\u8ddd\u79bb\u5c0f\u4e8e\u7b2c\u4e8c\u79cd\u5c31\u53ef\u4ee5\u66f4\u65b0 dis[x] > dis[t] + w; \u7ed9\u5b9a\u4e00\u4e2a n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe\uff0c\u56fe\u4e2d\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\uff0c\u6240\u6709\u8fb9\u6743\u5747\u4e3a\u6b63\u503c\u3002 \u8bf7\u4f60\u6c42\u51fa 1 \u53f7\u70b9\u5230 n \u53f7\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\uff0c\u5982\u679c\u65e0\u6cd5\u4ece 1 \u53f7\u70b9\u8d70\u5230 n \u53f7\u70b9\uff0c\u5219\u8f93\u51fa […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[72],"tags":[],"class_list":["post-6","post","type-post","status-publish","format-standard","hentry","category-acm"],"_links":{"self":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts\/6","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6"}],"version-history":[{"count":0,"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts\/6\/revisions"}],"wp:attachment":[{"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}