\n| \u50a8\u5b58u\u70b9\u5230\u8fbe\u7684\u6240\u6709v\u70b9\u53ca\u5176\u6743\u503c\u7684\u6570\u7ec4（u\u4e3a\u8d77\u59cb，v\u4e3a\u7ec8\u6b62）<\/td>\n | \u68c0\u67e5\u67d0\u4e2a\u70b9\u662f\u5426\u5b58\u5728\u961f\u5217<\/td>\n | \u8bb0\u5f55\u6bcf\u4e2a\u70b9\u8fdb\u5165\u961f\u5217\u7684\u6b21\u6570<\/td>\n | \u8bb0\u5f55\u524d\u9a71\u8282\u70b9<\/td>\n | \u8ba9\u53ef\u80fd\u662f\u6700\u77ed\u8def\u5f84\u7684\u70b9\u5165\u8fdb\u6765<\/td>\n | \u8bb0\u5f55\u6e90\u70b9\u8ddd<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u4e0b\u9762\u4e3e\u4f8b\u5b50\u6a21\u62df\u4e00\u4e0b：<\/p>\n \u5b8c\u6210\u521d\u59cb\u5316：<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214956-689bb704b8373.png\") <\/p>\n
\u5177\u4f53\u64cd\u4f5c\u5982\u56fe：<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214956-689bb704f3f82.png\") <\/p>\n
\u8fd9\u6837\u4e00\u6765\u5176\u5b9e\u86ee\u7b80\u5355\u7684\u5427；\u5f53\u7136\u4ee3\u7801\u4e5f\u662f\u5982\u6b64；\u6ce8\u610f\u4e00\u4e9b\u7ec6\u8282\u5c31\u597d\u5566！\u00a0<\/p>\n \u4e09.\u7b97\u6cd5\u8be6\u7ec6\u6b65\u9aa4:\u00a0<\/h2>\n\u4e0b\u9762\u6211\u4eec\u5206\u56db\u6b65\u6765\u5bf9\u4ee3\u7801\u8fdb\u884c\u5dee\u5206\u8fc7\u7a0b；\u8fdb\u884c\u76f8\u5173\u4ee3\u7801\u5256\u6790\u4e66\u5199\u64cd\u4f5c：<\/p>\n 3.1\u521d\u59cb\u5316\u64cd\u4f5c：<\/h2>\n\u4e0b\u9762\u6211\u4eec\u6839\u636e\u4e0a\u9762\u7684\u4f8b\u5b50\u53ca\u8868\u683c\u5b8c\u6210\u5bf9\u5e94\u4ee3\u7801\u4e66\u5199：<\/p>\n \u521d\u59cb\u5316\u90bb\u63a5\u8868：<\/p>\n cin >> n >> m;
\nfor (int i = 0; i < m; i++) {
\nint x, y, z;
\n\/\/\u5355\u53cc\u5411\u8fb9\u90fd\u53ef：
\ncin >> x >> y >> z;
\nv[x].emplace_back(y, z);
\n} <\/p>\n \u00a0\u521d\u59cb\u5316pre\u6570\u7ec4：<\/p>\n void init_pre() {<\/p>\n for (int i = 1; i <= n; i++)pre[i] = i;
\n} <\/p>\n \u521d\u59cb\u5316dist：<\/p>\n fill(dist + 1, dist + n + 1, maxi);\/\/\u521d\u59cb\u5316dist\u8981\u6ce8\u610f：\u6211\u4eec\u4ece1-n\u7684\u70b9\u7684\u7f16\u53f7；\u4e5f\u5c31\u662f\u521d\u59cb\u5316\u8fd9\u4e2a\u533a\u95f4（[\u8fed\u4ee3\u5668)） <\/p>\n \u5176\u4ed6；\u8fd9\u6837\u7c7b\u4f3c\u7684\u521d\u59cb\u5316\u548c\u6211\u4eec\u4e0a\u4e00\u7bc7\u7684Bellman-Ford\u7b97\u6cd5\u5dee\u4e0d\u591a；\u53ef\u4ee5\u53c2\u8003：<\/p>\n \u3010\u72c2\u70ed\u7b97\u6cd5\u7bc7\u3011\u63a2\u5bfb\u56fe\u8bba\u5e7d\u5f84：Bellman – Ford \u7b97\u6cd5\u7684\u6d6a\u6f2b\u5f81\u7a0b（\u901a\u4fd7\u6613\u61c2\u7248）-CSDN\u535a\u5ba2<\/p>\n<\/p>\n 3.2\u961f\u5217\u8fed\u4ee3\u677e\u5f1b\u9636\u6bb5：<\/h2>\n\u4e5f\u5c31\u662f\u8bf4\u53ea\u8981\u961f\u5217\u4e0d\u5b58\u5728；\u7136\u540e\u6211\u4eec\u5bf9\u5230\u8fbe\u8fd9\u4e2ap\u70b9\u7684\u8fb9\u8fdb\u884c\u677e\u5f1b\u6210\u529f\u4e86；\u5373\u6b64\u65f6\u7684\u8def\u5f84\u6709\u53ef\u80fd\u662f\u5230\u8fbep\u6216\u8005\u901a\u8fc7p\u5230\u8fbe\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u5f84\u3002 \u00a0 \u00a0 \u00a0 \u56e0\u6b64\u53ef\u80fd\u4f5c\u4e3a\u6700\u77ed\u8def\u5f84\u53ca\u5b83\u7684\u4e00\u90e8\u5206；\u6545\u5165\u961f\u5217；\u4f46\u662f\u8fd8\u662f\u4e0d\u592a\u5b8c\u7f8e；\u56e0\u6b64\u5c31\u8981\u7528\u5230\u4e0b\u9762\u7684\u4f18\u5316\u4e86\u3002<\/p>\n dist[s] = 0;
\nq.push(s);
\nwhile (!q.empty()) {
\nint cur = q.front();
\nq.pop();
\nfor (auto [p,w] : v[cur]) {<\/p>\n if (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nq.push(p);<\/p>\n }
\n}<\/p>\n }<\/p>\n 3.3\u63d0\u901f\u5c0f\u4f18\u5316：<\/h2>\n\u8bb0\u5f55\u662f\u5426\u5728\u961f\u5217\u4e2d\u9632\u6b62\u91cd\u590d\u64cd\u4f5c；\u56e0\u4e3a\u5982\u679c\u961f\u5217\u6709\u76f8\u540c\u7684\u70b9；\u90a3\u4e48\u5bf9\u4e00\u4e2a\u70b9\u64cd\u4f5c\u5b8c\u53ea\u53ef\u80fd\u6539\u53d8dist p；\u81ea\u8eab\u7684dist\u4e0d\u4f1a\u53d8；\u4e5f\u5c31\u662f\u5f53\u518d\u6b21\u904d\u5386\u5230\u4e0b\u4e00\u4e2a\u540c\u70b9\u7684\u65f6\u5019；\u4e0d\u4f1a\u8fdb\u884c\u4efb\u4f55\u64cd\u4f5c\u3002<\/p>\n \u6240\u4ee5\u53ea\u9700\u8981\u5728\u4e0a\u9762\u4ee3\u7801\u52a0\u4e0a\u68c0\u67e5\u6570\u7ec4\u5373\u53ef：<\/p>\n dist[s] = 0;
\nq.push(s);
\nin_q[s] = 1;
\nwhile (!q.empty()) {
\nint cur = q.front();
\nq.pop();
\nin_q[cur] = 0;
\nfor (auto [p,w] : v[cur]) {<\/p>\n if (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {\/\/ \u4e0d\u5728\u961f\u5217\u4e2d\u624d\u5165\u961f
\nq.push(p);
\nin_q[p] = 1;<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n 3.4\u68c0\u67e5\u662f\u5426\u5b58\u5728\u8d1f\u73af：<\/h2>\n\u8fd9\u91cc\u6211\u4eec\u5148\u653e\u7ed3\u8bba；\u5982\u679c\u4e00\u4e2a\u70b9\u88ab\u5165\u961f\u5217\u6b21\u6570\u8d85\u8fc7\u4e86n\u6b21；\u90a3\u4e48\u5b83\u4e00\u5b9a\u662f\u5b58\u5728\u8d1f\u73af\u7684（\u6b63\u73af\u62160\u73af\u53ef\u4ee5\u63d0\u524d\u627e\u5230\u6700\u77ed\u8def\u5f84；\u53bb\u73af\u65e0\u5f71\u54cd；\u6545\u4e0d\u53ef\u80fd\u53d1\u751f）：<\/p>\n dist[s] = 0;
\nq.push(s);
\nin_q[s] = 1;
\ncnt[s]++;
\nwhile (!q.empty()) {
\nint cur = q.front();
\nq.pop();
\nin_q[cur] = 0;
\nfor (auto [p,w] : v[cur]) {
\nif (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {\/\/ \u4e0d\u5728\u961f\u5217\u4e2d\u624d\u5165\u961f
\nq.push(p);
\nin_q[p] = 1;
\ncnt[p]++;
\nif (cnt[p] > n) return 0;\/\/\u5224\u65ad\u8d1f\u73af<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n \u56db.\u4ee3\u7801\u5b9e\u73b0：<\/h2>\n#include<iostream>
\n#include<vector>
\n#include<algorithm>
\n#include<map>
\n#include<queue>
\nusing namespace std;
\nint maxi = 0x3f3f3f3f;
\nconst int N = 1e6 + 1;
\nint dist[N], n, m;
\nint pre[N];\/\/\u524d\u9a71\u8282\u70b9
\nvector<pair<int,int>> v[N];\/\/\u90bb\u63a5\u8868：\u50a8\u5b58u\u53ca\u6240\u8fde\u63a5\u7684v\u548c\u4e4b\u95f4\u7684w
\nqueue<int>q;
\nbool in_q[N] = { 0 };\/\/\u8bb0\u5f55\u662f\u5426\u5728\u961f\u5217\u4e2d\u9632\u6b62\u91cd\u590d\u64cd\u4f5c；\u56e0\u4e3a\u5982\u679c\u961f\u5217\u6709\u76f8\u540c\u7684\u70b9；\u90a3\u4e48\u5bf9\u4e00\u4e2a\u70b9\u64cd\u4f5c\u5b8c\u53ea\u53ef\u80fd\u6539\u53d8dist p；\u81ea\u8eab\u7684dist
\n \/\/\u4e0d\u4f1a\u53d8；\u4e5f\u5c31\u662f\u5f53\u518d\u6b21\u904d\u5386\u5230\u4e0b\u4e00\u4e2a\u540c\u70b9\u7684\u65f6\u5019；\u4e0d\u4f1a\u8fdb\u884c\u4efb\u4f55\u64cd\u4f5c\u3002
\nint cnt[N] = { 0 };\/\/\u8bb0\u5f55\u8282\u70b9\u5165\u961f\u5217\u6b21\u6570；\u5224\u65ad\u662f\u5426\u51fa\u73b0\u8d1f\u73af
\nbool SPFA(int s,int n) {\/\/n\u4e2a\u70b9
\nfill(dist + 1, dist + n + 1, maxi);\/\/\u521d\u59cb\u5316dist\u8981\u6ce8\u610f：\u6211\u4eec\u4ece1-n\u7684\u70b9\u7684\u7f16\u53f7；\u4e5f\u5c31\u662f\u521d\u59cb\u5316\u8fd9\u4e2a\u533a\u95f4（[\u8fed\u4ee3\u5668)）
\ndist[s] = 0;
\nq.push(s);
\nin_q[s] = 1;
\ncnt[s]++;
\nwhile (!q.empty()) {
\nint cur = q.front();
\nq.pop();
\nin_q[cur] = 0;
\nfor (auto [p,w] : v[cur]) {
\n \/\/\u8fd9\u91cc\u4e3a\u4f55\u4f1a\u5165\u961f\u5217？
\n \/\/\u4e5f\u5c31\u662f\u8bf4\u53ea\u8981\u961f\u5217\u4e0d\u5b58\u5728；\u7136\u540e\u6211\u4eec\u5bf9\u5230\u8fbe\u8fd9\u4e2ap\u70b9\u7684\u8fb9\u8fdb\u884c\u677e\u5f1b\u6210\u529f\u4e86；\u5373\u6b64\u65f6\u7684\u8def\u5f84\u6709\u53ef\u80fd\u662f\u5230\u8fbep\u6216\u8005\u901a\u8fc7p\u5230\u8fbe\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u5f84
\n \/\/\u56e0\u6b64\u53ef\u80fd\u4f5c\u4e3a\u6700\u77ed\u8def\u5f84\u53ca\u5b83\u7684\u4e00\u90e8\u5206；\u6545\u5165\u961f\u5217\u3002
\nif (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {\/\/ \u4e0d\u5728\u961f\u5217\u4e2d\u624d\u5165\u961f
\nq.push(p);
\nin_q[p] = 1;
\ncnt[p]++;
\nif (cnt[p] > n) return 0;\/\/\u5224\u65ad\u8d1f\u73af<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n return 1;<\/p>\n }<\/p>\n void init_pre() {<\/p>\n for (int i = 1; i <= n; i++)pre[i] = i;
\n}
\nint getpath(int x) {
\npre[x] == x ? x : getpath(pre[x]);
\ncout << x << " ";
\nreturn x;
\n}<\/p>\n int main() {
\ncin >> n >> m;
\nfor (int i = 0; i < m; i++) {
\nint x, y, z;
\n\/\/\u5355\u53cc\u5411\u8fb9\u90fd\u53ef：
\ncin >> x >> y >> z;
\nv[x].emplace_back(y, z);
\n}
\ninit_pre();<\/p>\n bool flag=SPFA(1,n);
\nif (flag) {
\nif (dist[n] == maxi) cout << "unconnected" << endl;
\nelse {
\ncout << dist[n] << endl;
\ncout << "\u6700\u77ed\u8def\u5f84：" << endl;
\ngetpath(n);
\n}
\n}
\nelse {
\ncout << "exist negative circle " << endl;
\n}
\nreturn 0;<\/p>\n } <\/p>\n \u4e94\u00b7\u4ee3\u7801\u5b9e\u4f8b\u6d4b\u8bd5：<\/h2>\n\u4e0b\u9762\u6211\u4eec\u5c31\u6765\u7528\u51e0\u7ec4\u6570\u636e\u6d4b\u8bd5\u4e00\u4e0b\u4e0a\u8ff0\u7684\u4ee3\u7801：<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb7058427e.png\") <\/p>\n
\u2460\u770b\u4e00\u4e0b\u5230\u8fbe6\u53f7\u70b9\u6700\u77ed\u8def\u5f84\u662f\u4e0d\u662f1；\u8def\u9014\u662f\u4e0d\u662f1 2 5 6：<\/p>\n \u00a0![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb705a29a7.png\") <\/p>\n \u4e5f\u662f\u6210\u7acb\u7684\u3002<\/p>\n \u518d\u6d4b\u8bd5\u4e00\u4e0b：<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb705affad.png\") <\/p>\n
\u2461\u770b\u4e00\u4e0b\u6700\u7ec81\u52304\u6700\u77ed\u8def\u5f84\u662f\u4e0d\u662f-3 \u8def\u9014\u662f\u4e0d\u662f1 3 4：\u00a0<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb705c436f.png\") <\/p>\n
\u2462\u6d4b\u8bd5\u4e00\u4e0b\u8d1f\u73af：<\/p>\n \u00a0![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb705d6891.png\") <\/p>\n \u6700\u7ec8\u4e5f\u662f\u6210\u7acb\u7684；\u4f46\u662f\u6709\u4e9b\u60c5\u51b5SPFA\u7b97\u6cd5\u6709\u65f6\u5019\u4f1a\u8d85\u65f6；\u56e0\u6b64\u9009\u62e9\u7684\u65f6\u5019\u8fd8\u9700\u6ce8\u610f\u3002\u00a0<\/p>\n \u516d.\u5c0f\u7ec6\u8282\u53ca\u7591\u70b9\u5256\u6790：<\/h2>\n\u6211\u4eec\u4e0b\u9762\u4ece\u8fd9\u51e0\u4e2a\u53ef\u80fd\u4f1a\u6709\u7591\u60d1\u7684\u70b9\u6765\u5206\u6790\u4e00\u4e0b：<\/p>\n 6.1 \u4e3a\u4ec0\u4e48\u53ef\u4ee5\u901a\u8fc7\u961f\u5217\u8fdb\u884cBellman-Ford\u7b97\u6cd5\u7684\u4f18\u5316\u64cd\u4f5c：<\/h2>\n\u6211\u4eec\u4e0a\u7bc7\u8bb2\u8ff0\u4e86Bellman-Ford\u7b97\u6cd5\u7684\u539f\u7406：\u5bf9\u5168\u8fb9n-1\u6b21\u677e\u5f1b；\u4f46\u662f\u8fd9\u6837\u5c31\u7b97\u63d0\u524d\u7ed3\u675f\u4e5f\u80af\u5b9a\u4f1a\u6709\u7684\u8fb9\u867d\u7136\u8ba9\u5b83\u677e\u5f1b\u4f46\u662f\u4e5f\u4e0d\u4f1a\u66f4\u65b0dist\u503c；\u4e5f\u5c31\u662f\u91cd\u590d\u64cd\u4f5c\u3002<\/p>\n \u90a3\u4e48\u6211\u4eec\u8fd9\u91cc\u5c31\u53ef\u4ee5\u7528\u961f\u5217：\u6bd4\u5982\u67d0\u4e2a\u70b9u\u677e\u5f1b\u540edist\u66f4\u65b0\u4e86\u4e5f\u5c31\u662f\u5b83\u627e\u5230\u4e86\u6700\u77ed\u8def\u5f84\u6216\u8005\u5176\u4ed6\u70b9\u53ef\u4ee5\u501f\u52a9\u8fd9\u4e2a\u70b9u\u8fdb\u884c\u627e\u6700\u77ed\u8def\u5f84;\u56e0\u6b64\u628a\u5b83\u5165\u961f\u5217\u65b9\u4fbf\u627e\u5230\u53ef\u4ee5\u501f\u52a9\u5b83\u4f5c\u4e3a\u4e2d\u95f4\u70b9\u7684\u5176\u4ed6\u70b9v（\u53ef\u80fd\u5c31\u662f\u6700\u77ed\u8def\u5f84）\u3002<\/p>\n \u00a0\u8fd9\u91cc\u4e3a\u4f55\u4f1a\u5165\u961f\u5217？\u00a0 \u4e5f\u5c31\u662f\u8bf4\u53ea\u8981\u961f\u5217\u4e0d\u5b58\u5728；\u7136\u540e\u6211\u4eec\u5bf9\u5230\u8fbe\u8fd9\u4e2ap\u70b9\u7684\u8fb9\u8fdb\u884c\u677e\u5f1b\u6210\u529f\u4e86；\u5373\u6b64\u65f6\u7684\u8def\u5f84\u6709\u53ef\u80fd\u662f\u5230\u8fbep\u6216\u8005\u901a\u8fc7p\u5230\u8fbe\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u5f84\u56e0\u6b64\u53ef\u80fd\u4f5c\u4e3a\u6700\u77ed\u8def\u5f84\u53ca\u5b83\u7684\u4e00\u90e8\u5206；\u6545\u5165\u961f\u5217\u3002<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214957-689bb705e3703.png\") <\/p>\n
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SLF：\u5373\u63d2\u5165\u65f6\u5019\u505a\u4f18\u5316；LLL：\u9009\u62e9\u5143\u7d20\u5220\u9664\u961f\u5217\u65f6\u505a\u4f18\u5316\u3002<\/p>\n 8.1SLF（Small Label First）\u4f18\u5316：<\/h2>\n8.1.1\u4f18\u5316\u539f\u7406：<\/h2>\n\u5728\u6bcf\u6b21\u53d6\u51fa\u961f\u9996\u9876\u70b9\u65f6，LLL \u4f18\u5316\u7b56\u7565\u4f1a\u68c0\u67e5\u961f\u9996\u9876\u70b9\u7684\u8ddd\u79bb\u662f\u5426\u5927\u4e8e\u5f53\u524d\u5df2\u786e\u5b9a\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u7684\u5e73\u5747\u503c\u3002\u5982\u679c\u5927\u4e8e，\u5219\u5c06\u961f\u9996\u9876\u70b9\u79fb\u5230\u961f\u5c3e，\u4f18\u5148\u5904\u7406\u8ddd\u79bb\u8f83\u5c0f\u7684\u9876\u70b9\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u907f\u514d\u957f\u65f6\u95f4\u5904\u7406\u8ddd\u79bb\u8f83\u5927\u7684\u9876\u70b9，\u9632\u6b62\u7b97\u6cd5\u5728\u4e00\u4e9b\u8ddd\u79bb\u8f83\u5927\u4f46\u5bf9\u6700\u7ec8\u7ed3\u679c\u5f71\u54cd\u8f83\u5c0f\u7684\u9876\u70b9\u4e0a\u6d6a\u8d39\u8fc7\u591a\u65f6\u95f4，\u4ece\u800c\u63d0\u9ad8\u7b97\u6cd5\u7684\u6574\u4f53\u6548\u7387\u3002<\/p>\n 8.1.2\u5b9e\u73b0\u65b9\u5f0f：<\/h2>\n\u5728\u53d6\u51fa\u961f\u9996\u9876\u70b9\u65f6\u589e\u52a0\u5224\u65ad\u903b\u8f91，\u6839\u636e\u6761\u4ef6\u5c06\u9876\u70b9\u79fb\u5230\u961f\u5c3e\u3002\u5b9e\u73b0\u65f6\u9700\u8981\u989d\u5916\u8bb0\u5f55\u5df2\u786e\u5b9a\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\u4ee5\u53ca\u5df2\u5904\u7406\u7684\u9876\u70b9\u6570，\u4ee5\u4fbf\u8ba1\u7b97\u5e73\u5747\u503c\u3002<\/p>\n \u7b80\u5355\u6765\u8bf4\u5c31\u662f：\u8fd9\u91cc\u7528\u53cc\u7aef\u961f\u5217\u8fdb\u884c\u4f18\u5316；\u53ef\u4ee5\u66f4\u5feb\u7684\u786e\u5b9a\u67d0\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u63d0\u524d\u66f4\u65b0\u51fa\u5230\u67d0\u70b9\u7684\u53ef\u80fd\u662f\u7684\u6700\u77ed\u8def\u5f84；\u52a0\u5feb\u4e86\u627e\u5230\u771f\u6b63\u7684\u6700\u77ed\u8def\u5f84\u6216\u8005\u4ee5\u53ca\u627e\u5230\u5f53\u524d\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u4ee5\u53ca\u540e\u5e8f\u7684\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u7684\u901f\u5ea6\u3002<\/p>\n \u56e0\u6b64\u6211\u4eec\u53ea\u9700\u8981\u5728\u6e90\u4ee3\u7801\u7279\u5224\u4e00\u4e0b：\u5728\u5165\u961f\u5217\u7684\u65f6\u5019\u548c\u961f\u5934\u5143\u7d20\u6bd4\u8f83\u4e00\u4e0b；\u5982\u679cdist\u8981\u5c0f\u4e8e\u961f\u5934；\u90a3\u4e48\u5c31\u63d2\u961f\u5934\u5426\u5219\u63d2\u961f\u5c3e\u3002<\/p>\n 8.1.3 SLF\u4f18\u5316\u4ee3\u7801\u5b9e\u73b0：<\/h2>\n#include<iostream>
\n#include<vector>
\n#include<algorithm>
\n#include<map>
\n#include<queue>
\nusing namespace std;
\nint maxi = 0x3f3f3f3f;
\nconst int N = 1e6 + 1;
\nint dist[N], n, m;
\nint pre[N];\/\/\u524d\u9a71\u8282\u70b9
\nvector<pair<int, int>> v[N];
\ndeque<int>q;
\nbool in_q[N] = { 0 };
\nint cnt[N] = { 0 };\/\/\u8bb0\u5f55\u8282\u70b9\u5165\u961f\u5217\u6b21\u6570
\nbool SPFA(int s, int n) {\/\/n\u4e2a\u70b9
\nfill(dist + 1, dist + n + 1, maxi);\/\/\u521d\u59cb\u5316dist\u8981\u6ce8\u610f：\u6211\u4eec\u4ece1-n\u7684\u70b9\u7684\u7f16\u53f7；\u4e5f\u5c31\u662f\u521d\u59cb\u5316\u8fd9\u4e2a\u533a\u95f4（[\u8fed\u4ee3\u5668)）
\ndist[s] = 0;
\nq.push_back(s);
\nin_q[s] = 1;
\ncnt[s]++;
\nwhile (!q.empty()) {
\nint cur = q.front();
\nq.pop_front();
\nin_q[cur] = 0;
\nfor (auto [p, w] : v[cur]) {
\nif (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {
\n\/\/\u8fd9\u91cc\u7528\u53cc\u7aef\u961f\u5217\u8fdb\u884c\u4f18\u5316；\u53ef\u4ee5\u66f4\u5feb\u7684\u786e\u5b9a\u67d0\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u63d0\u524d\u66f4\u65b0\u51fa\u5230\u67d0\u70b9\u7684\u53ef\u80fd\u662f\u7684\u6700\u77ed\u8def\u5f84
\n\/\/\u52a0\u5feb\u4e86\u627e\u5230\u771f\u6b63\u7684\u6700\u77ed\u8def\u5f84\u6216\u8005\u4ee5\u53ca\u627e\u5230\u5f53\u524d\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u4ee5\u53ca\u540e\u5e8f\u7684\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u7684\u901f\u5ea6
\nif (!q.empty() && dist[p] < dist[q.front()]) {
\nq.push_front(p);
\n}
\nelse {
\nq.push_back(p);
\n}
\nin_q[p] = 1;
\ncnt[p]++;
\nif (cnt[p] > n) return 0;<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n return 1;<\/p>\n }<\/p>\n void init_pre() {<\/p>\n for (int i = 1; i <= n; i++)pre[i] = i;
\n}
\nint getpath(int x) {
\npre[x] == x ? x : getpath(pre[x]);
\ncout << x << " ";
\nreturn x;
\n}<\/p>\n int main() {
\ncin >> n >> m;
\nfor (int i = 0; i < m; i++) {
\nint x, y, z;
\n\/\/\u5355\u53cc\u5411\u8fb9\u90fd\u53ef：
\ncin >> x >> y >> z;
\nv[x].emplace_back(y, z);
\n}
\ninit_pre();<\/p>\n bool flag = SPFA(1, n);
\nif (flag) {
\nif (dist[n] == maxi) cout << "unconnected" << endl;
\nelse {
\ncout << dist[n] << endl;
\ncout << "\u6700\u77ed\u8def\u5f84：" << endl;
\ngetpath(n);
\n}
\n}
\nelse {
\ncout << "exist negative circle " << endl;
\n}
\nreturn 0;<\/p>\n }<\/p>\n 8.2LLL（Large Label Last）\u4f18\u5316：<\/h2>\n8.2.1\u4f18\u5316\u539f\u7406：<\/h2>\n\u5728\u6bcf\u6b21\u53d6\u51fa\u961f\u9996\u9876\u70b9\u65f6，LLL \u4f18\u5316\u7b56\u7565\u4f1a\u68c0\u67e5\u961f\u9996\u9876\u70b9\u7684\u8ddd\u79bb\u662f\u5426\u5927\u4e8e\u5f53\u524d\u5df2\u786e\u5b9a\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u7684\u5e73\u5747\u503c\u3002\u5982\u679c\u5927\u4e8e，\u5219\u5c06\u961f\u9996\u9876\u70b9\u79fb\u5230\u961f\u5c3e，\u4f18\u5148\u5904\u7406\u8ddd\u79bb\u8f83\u5c0f\u7684\u9876\u70b9\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u907f\u514d\u957f\u65f6\u95f4\u5904\u7406\u8ddd\u79bb\u8f83\u5927\u7684\u9876\u70b9，\u9632\u6b62\u7b97\u6cd5\u5728\u4e00\u4e9b\u8ddd\u79bb\u8f83\u5927\u4f46\u5bf9\u6700\u7ec8\u7ed3\u679c\u5f71\u54cd\u8f83\u5c0f\u7684\u9876\u70b9\u4e0a\u6d6a\u8d39\u8fc7\u591a\u65f6\u95f4，\u4ece\u800c\u63d0\u9ad8\u7b97\u6cd5\u7684\u6574\u4f53\u6548\u7387\u3002<\/p>\n 8.2.2\u5b9e\u73b0\u65b9\u5f0f：<\/h2>\n\u5728\u53d6\u51fa\u961f\u9996\u9876\u70b9\u65f6\u589e\u52a0\u5224\u65ad\u903b\u8f91，\u6839\u636e\u6761\u4ef6\u5c06\u9876\u70b9\u79fb\u5230\u961f\u5c3e\u3002\u5b9e\u73b0\u65f6\u9700\u8981\u989d\u5916\u8bb0\u5f55\u5df2\u786e\u5b9a\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\u4ee5\u53ca\u5df2\u5904\u7406\u7684\u9876\u70b9\u6570，\u4ee5\u4fbf\u8ba1\u7b97\u5e73\u5747\u503c\u3002<\/p>\n \u7b80\u5355\u6765\u8bf4\u5c31\u662f：\u5728\u7b26\u5408front\u5904\u7684\u8282\u70b9\u5927\u4e8e\u5e73\u5747\u503c\u7684\u65f6\u5019；\u4e00\u76f4\u628a\u5b83\u79fb\u52a8\u5230back\u5904；\u76f4\u5230\u53d1\u73b0\u5c0f\u4e8e\u4e86；\u505c\u6b62；\u65b9\u4fbf\u540e\u9762\u53d6\u5f97front\u7684\u65f6\u5019\u662f\u76f8\u5bf9\u5c0f\u7684dsit；\u540cslf\u4f18\u5316；\u5c3d\u53ef\u80fd\u5148\u64cd\u4f5c\u8f83\u5c0f\u7684dist\u7684\u70b9\u3002<\/p>\n \u4e5f\u5c31\u662f：\u6211\u4eec\u6bcf\u6b21\u9009\u62e9\u961f\u5934\u5143\u7d20\u65f6\u5019\u628a\u5b83\u7684dsit\u548c\u5168\u961f\u7684dist\u6bd4\u8f83\u5982\u679c\u5c0f\u5c31\u9000\u51fa\u9009\u961f\u5934\u5f97\u65f6\u5019\u5c31\u4f1a\u9009\u5b83；\u5426\u5219\u628a\u5b83\u5e72\u5230\u961f\u5c3e；\u518d\u6b21\u6bd4\u8f83\u961f\u5934\u548c\u5e73\u5747\u503c\u3002<\/p>\n 8.2.3LLL\u4f18\u5316\u4ee3\u7801\u5b9e\u73b0：<\/h2>\n<\/p>\n#include<iostream>
\n#include<vector>
\n#include<algorithm>
\n#include<map>
\n#include<queue>
\nusing namespace std;
\nint maxi = 0x3f3f3f3f;
\nconst int N = 1e6 + 1;
\nint dist[N], n, m;
\nint pre[N];\/\/\u524d\u9a71\u8282\u70b9
\nvector<pair<int, int>> v[N];
\ndeque<int>q;
\nbool in_q[N] = { 0 };
\nint cnt[N] = { 0 };\/\/\u8bb0\u5f55\u8282\u70b9\u5165\u961f\u5217\u6b21\u6570
\nbool SPFA(int s, int n) {\/\/n\u4e2a\u70b9
\nfill(dist + 1, dist + n + 1, maxi);\/\/\u521d\u59cb\u5316dist\u8981\u6ce8\u610f：\u6211\u4eec\u4ece1-n\u7684\u70b9\u7684\u7f16\u53f7；\u4e5f\u5c31\u662f\u521d\u59cb\u5316\u8fd9\u4e2a\u533a\u95f4（[\u8fed\u4ee3\u5668)）
\ndist[s] = 0;
\nq.push_back(s);
\nin_q[s] = 1;
\ncnt[s]++;
\nint sum_dist = 0; \/\/ \u961f\u5217\u4e2d\u6240\u6709\u8282\u70b9\u7684\u8ddd\u79bb\u603b\u548c
\nint queue_size = 1; \/\/ \u961f\u5217\u4e2d\u8282\u70b9\u7684\u6570\u91cf
\nwhile (!q.empty()) {
\n \/\/\/LLL\u4f18\u5316：\u5728\u7b26\u5408front\u5904\u7684\u8282\u70b9\u5927\u4e8e\u5e73\u5747\u503c\u7684\u65f6\u5019；\u4e00\u76f4\u628a\u5b83\u79fb\u52a8\u5230back\u5904；\u76f4\u5230\u53d1\u73b0\u5c0f\u4e8e\u4e86；\u505c\u6b62；\u65b9\u4fbf\u540e\u9762\u53d6\u5f97front\u7684\u65f6\u5019\u662f\u76f8\u5bf9\u5c0f\u7684dsit
\n \/\/\/\u540cslf\u4f18\u5316；\u5c3d\u53ef\u80fd\u5148\u64cd\u4f5c\u8f83\u5c0f\u7684dist\u7684\u70b9
\nwhile (!q.empty() && dist[q.front()] * queue_size > sum_dist) {
\nint front_node = q.front();
\nq.pop_front();
\nq.push_back(front_node);
\n}<\/p>\n int cur = q.front();
\nq.pop_front();
\nin_q[cur] = 0;
\nsum_dist -= dist[cur];
\nqueue_size–;<\/p>\n for (auto [p, w] : v[cur]) {
\nif (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {
\nq.push_back(p);
\nin_q[p] = 1;
\ncnt[p]++;
\nsum_dist += dist[p];
\nqueue_size++;<\/p>\n if (cnt[p] > n) return 0;<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n return 1;<\/p>\n }<\/p>\n void init_pre() {<\/p>\n for (int i = 1; i <= n; i++)pre[i] = i;
\n}
\nint getpath(int x) {
\npre[x] == x ? x : getpath(pre[x]);
\ncout << x << " ";
\nreturn x;
\n}<\/p>\n int main() {
\ncin >> n >> m;
\nfor (int i = 0; i < m; i++) {
\nint x, y, z;
\n\/\/\u5355\u53cc\u5411\u8fb9\u90fd\u53ef：
\ncin >> x >> y >> z;
\nv[x].emplace_back(y, z);
\n}
\ninit_pre();<\/p>\n bool flag = SPFA(1, n);
\nif (flag) {
\nif (dist[n] == maxi) cout << "unconnected" << endl;
\nelse {
\ncout << dist[n] << endl;
\ncout << "\u6700\u77ed\u8def\u5f84：" << endl;
\ngetpath(n);
\n}
\n}
\nelse {
\ncout << "exist negative circle " << endl;
\n}
\nreturn 0;<\/p>\n }<\/p>\n LLL+SLF\u7684deque\u4f18\u5316：<\/p>\n 8.3SLF\u4e0eLLL\u8054\u5408\u4f18\u5316：<\/h2>\n8.3.1\u4f18\u5316\u539f\u7406：<\/h2>\n\u5c06 SLF \u548c LLL \u4e24\u79cd\u4f18\u5316\u65b9\u6cd5\u7ed3\u5408\u4f7f\u7528，\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u5728\u5404\u79cd\u56fe\u7ed3\u6784\u4e0b\u7684\u6027\u80fd\u3002SLF \u7b56\u7565\u5728\u5165\u961f\u65f6\u4f18\u5148\u5904\u7406\u8ddd\u79bb\u5c0f\u7684\u9876\u70b9，\u800c LLL \u7b56\u7565\u5728\u51fa\u961f\u65f6\u907f\u514d\u957f\u65f6\u95f4\u5904\u7406\u8ddd\u79bb\u5927\u7684\u9876\u70b9，\u4e24\u8005\u76f8\u4e92\u914d\u5408，\u80fd\u66f4\u6709\u6548\u5730\u51cf\u5c11\u4e0d\u5fc5\u8981\u7684\u8ba1\u7b97，\u52a0\u5feb\u7b97\u6cd5\u6536\u655b\u3002<\/p>\n 8.3.2\u5b9e\u73b0\u65b9\u5f0f：<\/h2>\n\u5728\u4ee3\u7801\u4e2d\u540c\u65f6\u5b9e\u73b0 SLF \u548c LLL \u7684\u903b\u8f91\u3002\u5728\u5165\u961f\u65f6\u91c7\u7528 SLF \u7684\u5165\u961f\u65b9\u5f0f，\u5728\u51fa\u961f\u65f6\u91c7\u7528 LLL \u7684\u5224\u65ad\u548c\u5904\u7406\u65b9\u5f0f，\u901a\u8fc7\u5408\u7406\u7684\u903b\u8f91\u7ec4\u5408，\u5145\u5206\u53d1\u6325\u4e24\u79cd\u4f18\u5316\u7b56\u7565\u7684\u4f18\u52bf\u3002<\/p>\n \u8fd8\u662f\u7b80\u5355\u8bf4\u4e00\u4e0b：\u6211\u4eec\u7684LLL\u548cSLF\u4f18\u5316\u90fd\u662f\u8ba9\u6bcf\u6b21\u9009\u62e9\u961f\u5934\u5143\u7d20\u7684\u65f6\u5019\u64cd\u4f5c\u7684\u5c3d\u91cf\u662f\u6e90\u70b9\u8ddd\u5c0f\u7684\u70b9；\u90a3\u4e48\u5b83\u6709\u53ef\u80fd\u5c31\u662f\u6700\u77ed\u6e90\u70b9\u8ddd；\u6216\u8005\u5176\u4ed6\u70b9\u53ef\u4ee5\u501f\u52a9\u5b83\u5b9e\u73b0\u6700\u77ed\u6e90\u70b9\u8ddd\u7684\u67e5\u627e\u4ece\u800c\u66f4\u5feb\u7684\u627e\u5230\u5bf9\u5e94\u7684\u6700\u7ec8\u7684dist\u503c\u3002<\/p>\n \u5f52\u6839\u7ed3\u5e95：\u5c31\u662f\u4e3a\u4e86\u63d0\u524d\u64cd\u4f5c\u6e90\u70b9\u8ddd\u66f4\u5c0f\u7684\u70b9；\u8ba9\u6bcf\u4e2a\u70b9\u7684\u6700\u7ec8dist\u503c\u5c3d\u5feb\u66f4\u65b0\u5b8c；\u51cf\u5c11\u70b9\u5165\u961f\u5217\u7684\u6b21\u6570，\u63d0\u524d\u7ec8\u6b62\u5165\u51fa\u961f\u5217\u64cd\u4f5c\u3002<\/p>\n 8.3.3SLF\u4e0eLLL\u8054\u5408\u4f18\u5316\u4ee3\u7801\u5b9e\u73b0：<\/h2>\n#include<iostream>
\n#include<vector>
\n#include<algorithm>
\n#include<map>
\n#include<queue>
\nusing namespace std;
\nint maxi = 0x3f3f3f3f;
\nconst int N = 1e6 + 1;
\nint dist[N], n, m;
\nint pre[N];\/\/\u524d\u9a71\u8282\u70b9
\nvector<pair<int, int>> v[N];
\ndeque<int>q;
\nbool in_q[N] = { 0 };
\nint cnt[N] = { 0 };\/\/\u8bb0\u5f55\u8282\u70b9\u5165\u961f\u5217\u6b21\u6570
\nbool SPFA(int s, int n) {\/\/n\u4e2a\u70b9
\nfill(dist + 1, dist + n + 1, maxi);\/\/\u521d\u59cb\u5316dist\u8981\u6ce8\u610f：\u6211\u4eec\u4ece1-n\u7684\u70b9\u7684\u7f16\u53f7；\u4e5f\u5c31\u662f\u521d\u59cb\u5316\u8fd9\u4e2a\u533a\u95f4（[\u8fed\u4ee3\u5668)）
\ndist[s] = 0;
\nq.push_back(s);
\nin_q[s] = 1;
\ncnt[s]++;
\nint sum_dist = 0; \/\/ \u961f\u5217\u4e2d\u6240\u6709\u8282\u70b9\u7684\u8ddd\u79bb\u603b\u548c
\nint queue_size = 1; \/\/ \u961f\u5217\u4e2d\u8282\u70b9\u7684\u6570\u91cf
\nwhile (!q.empty()) {
\n\/\/\/LLL\u4f18\u5316：\u5728\u7b26\u5408front\u5904\u7684\u8282\u70b9\u5927\u4e8e\u5e73\u5747\u503c\u7684\u65f6\u5019；\u4e00\u76f4\u628a\u5b83\u79fb\u52a8\u5230back\u5904；\u76f4\u5230\u53d1\u73b0\u5c0f\u4e8e\u4e86；\u505c\u6b62；\u65b9\u4fbf\u540e\u9762\u53d6\u5f97front\u7684\u65f6\u5019\u662f\u76f8\u5bf9\u5c0f\u7684dsit
\n\/\/ \/\/\/\u540cslf\u4f18\u5316；\u5c3d\u53ef\u80fd\u5148\u64cd\u4f5c\u8f83\u5c0f\u7684dist\u7684\u70b9
\nwhile (!q.empty() && dist[q.front()] * queue_size > sum_dist) {
\nint front_node = q.front();
\nq.pop_front();
\nq.push_back(front_node);
\n}<\/p>\n int cur = q.front();
\nq.pop_front();
\nin_q[cur] = 0;
\nsum_dist -= dist[cur];
\nqueue_size–;<\/p>\n for (auto [p, w] : v[cur]) {
\nif (dist[cur] != maxi && dist[cur] + w < dist[p]) {
\ndist[p] = dist[cur] + w;
\npre[p] = cur;
\nif (!in_q[p]) {
\n\/\/\u8fd9\u91cc\u7528\u53cc\u7aef\u961f\u5217\u8fdb\u884c\u4f18\u5316；\u53ef\u4ee5\u66f4\u5feb\u7684\u786e\u5b9a\u67d0\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u63d0\u524d\u66f4\u65b0\u51fa\u5230\u67d0\u70b9\u7684\u53ef\u80fd\u662f\u7684\u6700\u77ed\u8def\u5f84
\n\/\/\/\/\u52a0\u5feb\u4e86\u627e\u5230\u771f\u6b63\u7684\u6700\u77ed\u8def\u5f84\u6216\u8005\u4ee5\u53ca\u627e\u5230\u5f53\u524d\u70b9\u7684\u6700\u77ed\u8def\u5f84；\u4ee5\u53ca\u540e\u5e8f\u7684\u5176\u4ed6\u70b9\u7684\u6700\u77ed\u8def\u7684\u901f\u5ea6
\nif (!q.empty() && dist[p] < dist[q.front()]) {
\nq.push_front(p);
\n}
\nelse {
\nq.push_back(p);
\n}
\nin_q[p] = 1;
\ncnt[p]++;
\nsum_dist += dist[p];
\nqueue_size++;<\/p>\n if (cnt[p] > n) return 0;<\/p>\n }
\n}
\n}<\/p>\n }<\/p>\n return 1;<\/p>\n }<\/p>\n void init_pre() {<\/p>\n for (int i = 1; i <= n; i++)pre[i] = i;
\n}
\nint getpath(int x) {
\npre[x] == x ? x : getpath(pre[x]);
\ncout << x << " ";
\nreturn x;
\n}<\/p>\n int main() {
\ncin >> n >> m;
\nfor (int i = 0; i < m; i++) {
\nint x, y, z;
\n\/\/\u5355\u53cc\u5411\u8fb9\u90fd\u53ef：
\ncin >> x >> y >> z;
\nv[x].emplace_back(y, z);
\n}
\ninit_pre();<\/p>\n bool flag = SPFA(1, n);
\nif (flag) {
\nif (dist[n] == maxi) cout << "unconnected" << endl;
\nelse {
\ncout << dist[n] << endl;
\ncout << "\u6700\u77ed\u8def\u5f84：" << endl;
\ngetpath(n);
\n}
\n}
\nelse {
\ncout << "exist negative circle " << endl;
\n}
\nreturn 0;<\/p>\n }<\/p>\n \u4e5d.\u7b97\u6cd5\u4f18\u52bf\u4e0e\u5c40\u9650\u6027:<\/h2>\n9.1\u4f18\u52bf：<\/h4>\n\u53ef\u5904\u7406\u8d1f\u6743\u8fb9：\u80fd\u5e94\u5bf9\u5305\u542b\u8d1f\u6743\u8fb9\u7684\u56fe，\u50cf\u4ea4\u901a\u7f51\u7edc\u4e2d\u8fd0\u8f93\u8865\u8d34\u60c5\u51b5，\u53ef\u51c6\u786e\u8ba1\u7b97\u6700\u4f18\u8def\u5f84\u3002<\/li>\n\u5e73\u5747\u590d\u6742\u5ea6\u4f4e：\u501f\u52a9\u961f\u5217\u4f18\u5316，\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(km) ，\u5728\u7a00\u758f\u56fe\u4e2d\u6536\u655b\u5feb\u3002<\/li>\n\u80fd\u68c0\u6d4b\u8d1f\u6743\u73af：\u901a\u8fc7\u8bb0\u5f55\u9876\u70b9\u5165\u961f\u6b21\u6570\u5224\u65ad\u8d1f\u6743\u73af，\u52a9\u4e8e\u91d1\u878d\u5957\u5229\u7b49\u573a\u666f\u51b3\u7b56\u3002<\/li>\n\u5b9e\u73b0\u7b80\u5355：\u57fa\u4e8e\u961f\u5217\u548c\u677e\u5f1b\u64cd\u4f5c，\u4ee3\u7801\u7ed3\u6784\u6e05\u6670，\u9002\u5408\u521d\u5b66\u8005\u548c\u5feb\u901f\u5b9e\u73b0\u9700\u6c42\u3002<\/li>\n9.2\u5c40\u9650\u6027:<\/h4>\n\u6700\u574f\u590d\u6742\u5ea6\u9ad8：\u7279\u6b8a\u56fe\u7ed3\u6784\u4e0b，\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u4e3a O(nm)，\u8fd0\u884c\u65f6\u95f4\u957f\u3002<\/li>\n\u6613\u88ab\u7279\u6b8a\u56fe\u5361\u65f6：\u5982\u5b8c\u5168\u56fe\u4f1a\u4f7f\u7b97\u6cd5\u9891\u7e41\u5165\u961f\u51fa\u961f，\u964d\u4f4e\u6027\u80fd\u3002<\/li>\n\u7a7a\u95f4\u5f00\u9500\u5927：\u9700\u961f\u5217\u53ca\u591a\u4e2a\u6570\u7ec4，\u5927\u89c4\u6a21\u56fe\u6570\u636e\u53ef\u80fd\u5185\u5b58\u4e0d\u8db3\u3002<\/li>\n\u4e0d\u9002\u7528\u4e8e\u7a20\u5bc6\u56fe：\u7a20\u5bc6\u56fe\u4e2d，\u961f\u5217\u548c\u677e\u5f1b\u64cd\u4f5c\u591a，\u6548\u7387\u4e0d\u5982\u4e13\u95e8\u7b97\u6cd5\u3002<\/li>\n\u5341\u00b7\u5e94\u7528\u573a\u666f：<\/h2>\n10.1\u4ea4\u901a\u7f51\u7edc\u89c4\u5212：<\/h2>\n\u2460\u53ef\u5904\u7406\u8d1f\u6743\u8fb9，\u80fd\u5e94\u5bf9\u9053\u8def\u56e0\u7279\u6b8a\u6d3b\u52a8（\u5982\u4fc3\u9500\u6d3b\u52a8\u5e26\u6765\u8fd0\u8f93\u8865\u8d34）\u5bfc\u81f4\u6210\u672c\u4e3a\u8d1f\u7684\u60c5\u51b5，\u89c4\u5212\u51fa\u6700\u4f18\u8fd0\u8f93\u8def\u5f84\u3002<\/p>\n \u2461\u5e73\u5747\u590d\u6742\u5ea6\u4f4e，\u5728\u57ce\u5e02\u4ea4\u901a\u7b49\u7a00\u758f\u56fe\u7ed3\u6784\u4e2d，\u80fd\u5feb\u901f\u7b97\u51fa\u4e24\u70b9\u95f4\u7684\u6700\u77ed\u51fa\u884c\u8def\u7ebf\u3002<\/p>\n 10.2\u91d1\u878d\u5957\u5229\u5206\u6790：<\/h2>\n\u80fd\u68c0\u6d4b\u8d1f\u6743\u73af，\u53ef\u53d1\u73b0\u91d1\u878d\u5e02\u573a\u4e2d\u56e0\u6c47\u7387\u5dee\u5f02\u3001\u5229\u7387\u6ce2\u52a8\u7b49\u56e0\u7d20\u5f62\u6210\u7684\u5957\u5229\u673a\u4f1a，\u8f85\u52a9\u6295\u8d44\u8005\u51b3\u7b56\u3002<\/p>\n 10.3\u793e\u4ea4\u7f51\u7edc\u4fe1\u606f\u4f20\u64ad：<\/h2>\n\u5e73\u5747\u590d\u6742\u5ea6\u4f4e，\u9002\u5408\u793e\u4ea4\u7f51\u7edc\u8fd9\u79cd\u7a00\u758f\u56fe，\u53ef\u8ba1\u7b97\u4fe1\u606f\u4ece\u4e00\u4e2a\u7528\u6237\u5230\u5176\u4ed6\u7528\u6237\u7684\u6700\u77ed\u4f20\u64ad\u8def\u5f84，\u52a9\u529b\u7cbe\u51c6\u8425\u9500\u548c\u4fe1\u606f\u6269\u6563\u7b56\u7565\u5236\u5b9a\u3002<\/p>\n 10.4\u7ade\u8d5b\u4e0e\u5b66\u4e60：<\/h2>\n\u5b9e\u73b0\u7b80\u5355，\u4fbf\u4e8e\u521d\u5b66\u8005\u7406\u89e3\u548c\u5feb\u901f\u5b9e\u73b0，\u5e38\u7528\u4e8e\u7b97\u6cd5\u7ade\u8d5b\u548c\u5b66\u4e60\u4e2d\u9a8c\u8bc1\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u3002<\/p>\n \u8fd8\u6709\u5176\u4ed6\u7684\u7528\u9014\u7b49\u7b49；\u8fd9\u91cc\u5c31\u4e0d\u4e00\u4e00\u5217\u4e3e\u4e86\u3002\u00a0<\/p>\n \u4f46\u662f\u8fd8\u6709\u4e9b\u7531\u4e8e\u5b83\u7684\u7f3a\u70b9\u662f\u4e0d\u80fd\u5e94\u7528\u7684；\u5982：<\/p>\n \u5927\u89c4\u6a21\u7a20\u5bc6\u7f51\u7edc：\u5982\u96c6\u6210\u7535\u8def\u5e03\u7ebf\u95ee\u9898，SPFA \u53ef\u80fd\u56e0\u6700\u574f\u590d\u6742\u5ea6\u9ad8\u3001\u4e0d\u9002\u7528\u4e8e\u7a20\u5bc6\u56fe\u800c\u6548\u7387\u4f4e\u4e0b，\u53ef\u8003\u8651\u4f7f\u7528\u9488\u5bf9\u7a20\u5bc6\u56fe\u4f18\u5316\u7684\u7b97\u6cd5，\u5982\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684 Dijkstra \u7b97\u6cd5\u53d8\u79cd\u3002<\/li>\n\u8d85\u5927\u89c4\u6a21\u56fe\u6570\u636e：\u5728\u5168\u7403\u4e92\u8054\u7f51\u62d3\u6251\u56fe\u5206\u6790\u4e2d，\u56e0\u7a7a\u95f4\u5f00\u9500\u5927\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u4e0d\u8db3，\u53ef\u8003\u8651\u91c7\u7528\u5206\u5e03\u5f0f\u7b97\u6cd5\u6216\u5185\u5b58\u4f18\u5316\u7b56\u7565\u3002<\/li>\n\u5341\u4e00\u00b7\u4e2a\u4eba\u5c0f\u603b\u7ed3：<\/h2>\n\u672c\u7bc7\u6211\u4eec\u6240\u4ecb\u7ecd\u7684\u662fSPFA\u7b97\u6cd5\u4e5f\u5c31\u662f\u4f18\u5316\u540e\u7684Bellman-Ford\u7b97\u6cd5；\u4e0b\u9762\u535a\u4e3b\u603b\u7ed3\u4e86\u4e00\u4e9b\u4ee3\u7801\u5b9e\u73b0\u6280\u5de7\u53ca\u5176\u9002\u7528\u7684\u5f52\u7eb3；\u8fd8\u671b\u5bf9\u5927\u5bb6\u5b66\u4e60\u8fd9\u4e2a\u7b97\u6cd5\u6709\u5e2e\u52a9\u3002<\/p>\n \u5b9e\u73b0SPFA：\u901a\u8fc7\u90bb\u63a5\u8868u\u8fde\u7740\u591a\u4e2av\u4ee5\u53caw（\u6743\u8fb9）；\u53ea\u8981v\u80fd\u501f\u52a9\u8fd9\u4e2au\u5230\u8fbe\u4e14\u66f4\u65b0dist（\u6ce8\u610fu\u70b9\u8981\u662f\u6e90\u70b9\u53ef\u8fbe\u70b9）\u5e76\u4e14v\u4e0d\u5728\u961f\u5217\u5c31\u5165\u8fdb\u53bb；\u65f6\u523b\u6807\u8bb0\u662f\u5426\u5728\u961f\u5217\u60c5\u51b5\u4ee5\u53ca\u67d0\u70b9\u8fdb\u8fc7\u961f\u5217\u7684\u6b21\u6570cnt\u3002（\u9700\u8981\u65f6\u53ef\u7ef4\u62a4\u597dpre\u6570\u7ec4） \u8fd9\u4e5f\u5c31\u662f\u666e\u901aSPFA\u5b9e\u73b0\u3002<\/p>\n<\/p>\n \u5bf9\u5e94SLF\u548cLLL\u4f18\u5316：\u5176\u5b9e\u4e00\u4e2a\u662f\u63d2\u5165\u961f\u5217\u7684\u65f6\u5019：\u5c0f\u4e8e\u961f\u5934\u5143\u7d20\u5c31\u63d2\u961f\u5934\u5927\u4e8e\u5c31\u63d2\u961f\u5c3e\u3002<\/p>\n \u53e6\u4e00\u4e2a\u5c31\u662f\u9009\u62e9\u961f\u5217\u4e2d\u5143\u7d20\u7684\u65f6\u5019，\u6211\u4eec\u6bcf\u6b21\u90fd\u8981\u9009\u5bf9\u5934；\u56e0\u6b64\u5982\u679c\u53d1\u73b0\u961f\u5934\u5143\u7d20\u7684dist\u5c0f\u4e8e\u5e73\u5747\u503c\u5c31\u653e\u961f\u5c3e；\u5426\u5219\u9000\u51fa\u5faa\u73af\u9009\u62e9\u5b83（\u8fd9\u6837\u5c31\u5b9e\u73b0\u4e86\u6bcf\u6b21\u64cd\u4f5c\u961f\u5217\u4e2d\u5c3d\u53ef\u80fddist\u5c0f\u7684\u70b9）<\/p>\n<\/p>\n \u56db\u79cd\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u5c0f\u603b\u7ed3：<\/p>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250812214958-689bb706ead25.png\") <\/p>\n<\/p>\n
\u00a0\u8fd9\u7bc7\u6211\u4eec\u7684SPFA\u7b97\u6cd5\u5c31\u5206\u4eab\u5230\u6b64；\u4e0b\u4e00\u7bc7\u535a\u4e3b\u5c06\u5e26\u5927\u5bb6\u5256\u6790\u5176\u4ed6\u56fe\u8bba\u7b97\u6cd5；\u6b22\u8fce\u5927\u5bb6\u8ba2\u9605\u5440！！！<\/p><\/p>\n","protected":false},"excerpt":{"rendered":" \u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb1.8k\u6b21\uff0c\u70b9\u8d5e122\u6b21\uff0c\u6536\u85cf78\u6b21\u3002\u5728\u7b97\u6cd5\u7684\u5947\u5999\u5b87\u5b99\u4e2d\uff0cSPFA\uff08Shortest Path Faster Algorithm\uff09\u662f\u4e00\u9897\u72ec\u7279\u7684\u661f\u8fb0\u3002\u5b83\u6253\u7834\u5e38\u89c4\uff0c\u5728\u56fe\u7684\u590d\u6742\u7f51\u7edc\u91cc\uff0c\u4ee5\u7b80\u6d01\u9ad8\u6548\u7684\u65b9\u5f0f\uff0c\u5feb\u901f\u7a7f\u68ad\u4e8e\u5404\u4e2a\u8282\u70b9\uff0c\u7cbe\u51c6\u5b9a\u4f4d\u6700\u77ed\u8def\u5f84\u3002\u51ed\u501f\u677e\u5f1b\u64cd\u4f5c\u8fd9\u4e00\u795e\u5947 \u201c\u9b54\u6cd5\u201d\uff0c\u6bcf\u6b21\u8fed\u4ee3\u90fd\u5411\u7740\u6700\u4f18\u89e3\u5927\u6b65\u8fc8\u8fdb\uff0c\u8ba9\u590d\u6742\u7684\u8def\u5f84\u89c4\u5212\u96be\u9898\u8fce\u5203\u800c\u89e3 \u3002<\/p>\n","protected":false},"author":2,"featured_media":54412,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,5544,1984,5542,427,5543],"topic":[],"class_list":["post-54426","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-c","tag-spfa","tag-1984","tag-5542","tag-427","tag-5543"],"yoast_head":"\n \u3010\u72c2\u70ed\u7b97\u6cd5\u7bc7\u3011\u63a2\u5bfb\u56fe\u8bba\u5e7d\u5f84\u4e4bSPFA\u7b97\u6cd5\uff1a\u56fe\u8bba\u8ff7\u5bab\u91cc\u7684\u95ea\u7535\u5bfb\u5f84\u8005(\u901a\u4fd7\u6613\u61c2\u7248) - 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