{"id":56555,"date":"2025-08-14T20:15:44","date_gmt":"2025-08-14T12:15:44","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/56555.html"},"modified":"2025-08-14T20:15:44","modified_gmt":"2025-08-14T12:15:44","slug":"%e3%80%8a%e7%ae%97%e6%b3%95%e5%af%bc%e8%ae%ba%e3%80%8b%e7%ac%ac-24-%e7%ab%a0-%e5%8d%95%e6%ba%90%e6%9c%80%e7%9f%ad%e8%b7%af%e5%be%84","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/56555.html","title":{"rendered":"\u300a\u7b97\u6cd5\u5bfc\u8bba\u300b\u7b2c 24 \u7ae0 - \u5355\u6e90\u6700\u77ed\u8def\u5f84"},"content":{"rendered":"

\"\"<\/h2>\n

\u5f15\u8a00<\/span><\/h3>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u6838\u5fc3\u95ee\u9898\u4e4b\u4e00，\u65e8\u5728\u6c42\u89e3\u4ece\u56fe\u4e2d\u4e00\u4e2a\u7279\u5b9a\u6e90\u8282\u70b9\u5230\u5176\u4ed6\u6240\u6709\u8282\u70b9\u7684\u6700\u77ed\u8def\u5f84<\/span>\u3002\u8fd9\u4e00\u95ee\u9898\u5728\u5bfc\u822a\u7cfb\u7edf\u3001\u7f51\u7edc\u8def\u7531\u3001\u4ea4\u901a\u89c4\u5212\u7b49\u9886\u57df\u6709\u7740\u5e7f\u6cdb\u5e94\u7528\u3002\u672c\u7ae0\u5c06\u8be6\u7ec6\u89e3\u6790\u300a\u7b97\u6cd5\u5bfc\u8bba\u300b\u7b2c 24 \u7ae0\u4ecb\u7ecd\u7684\u56db\u79cd\u7ecf\u5178\u7b97\u6cd5，\u5e76\u63d0\u4f9b\u5b8c\u6574\u53ef\u8fd0\u884c\u7684 C++ \u5b9e\u73b0\u4ee3\u7801\u3002<\/p>\n

<\/h4>\n

24.1 Bellman-Ford \u7b97\u6cd5<\/span><\/h3>\n

\"\"<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Bellman-Ford \u7b97\u6cd5\u662f\u6c42\u89e3\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u7ecf\u5178\u7b97\u6cd5，\u5b83\u80fd\u591f\u5904\u7406\u5305\u542b\u8d1f\u6743\u8fb9\u7684\u56fe，\u5e76\u4e14\u53ef\u4ee5\u68c0\u6d4b\u51fa\u56fe\u4e2d\u662f\u5426\u5b58\u5728\u4ece\u6e90\u8282\u70b9\u53ef\u8fbe\u7684\u8d1f\u6743\u56de\u8def<\/span>\u3002<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/span><\/h4>\n
  • \u521d\u59cb\u5316：\u5c06\u6e90\u8282\u70b9\u5230\u6240\u6709\u5176\u4ed6\u8282\u70b9\u7684\u8ddd\u79bb\u521d\u59cb\u5316\u4e3a\u65e0\u7a77\u5927，\u6e90\u8282\u70b9\u5230\u81ea\u8eab\u7684\u8ddd\u79bb\u521d\u59cb\u5316\u4e3a 0\u3002<\/li>\n
  • \u677e\u5f1b\u64cd\u4f5c：\u5bf9\u56fe\u4e2d\u6240\u6709\u8fb9\u8fdb\u884c V-1 \u6b21\u677e\u5f1b\u64cd\u4f5c（V \u4e3a\u8282\u70b9\u6570）\u3002\u6bcf\u6b21\u677e\u5f1b\u64cd\u4f5c\u904d\u5386\u6240\u6709\u8fb9 (u, v)，\u82e5distance[v] > distance[u] + weight(u, v)，\u5219\u66f4\u65b0distance[v] = distance[u] + weight(u, v)\u3002<\/li>\n
  • \u68c0\u6d4b\u8d1f\u6743\u56de\u8def：\u5728\u5b8c\u6210 V-1 \u6b21\u677e\u5f1b\u540e，\u518d\u5bf9\u6240\u6709\u8fb9\u8fdb\u884c\u4e00\u6b21\u68c0\u67e5\u3002\u5982\u679c\u4ecd\u80fd\u8fdb\u884c\u677e\u5f1b\u64cd\u4f5c，\u5219\u8bf4\u660e\u56fe\u4e2d\u5b58\u5728\u4ece\u6e90\u8282\u70b9\u53ef\u8fbe\u7684\u8d1f\u6743\u56de\u8def\u3002<\/li>\n

    \"\"<\/h4>\n

    \u4ee3\u7801\u5b9e\u73b0<\/span><\/h4>\n

    #include <iostream>
    \n#include <vector>
    \n#include <climits>
    \nusing namespace std;<\/p>\n

    \/\/ \u8fb9\u7684\u7ed3\u6784\u4f53
    \nstruct Edge {
    \n int u; \/\/ \u8d77\u70b9
    \n int v; \/\/ \u7ec8\u70b9
    \n int weight; \/\/ \u6743\u91cd
    \n Edge(int u_, int v_, int weight_) : u(u_), v(v_), weight(weight_) {}
    \n};<\/p>\n

    \/**
    \n * Bellman-Ford\u7b97\u6cd5\u5b9e\u73b0
    \n * @param edges \u8fb9\u7684\u96c6\u5408
    \n * @param V \u8282\u70b9\u6570\u91cf
    \n * @param s \u6e90\u8282\u70b9
    \n * @param distance \u5b58\u50a8\u6700\u77ed\u8ddd\u79bb\u7684\u6570\u7ec4
    \n * @return \u662f\u5426\u5b58\u5728\u8d1f\u6743\u56de\u8def
    \n *\/
    \nbool bellmanFord(const vector<Edge>& edges, int V, int s, vector<int>& distance) {
    \n \/\/ 1. \u521d\u59cb\u5316\u8ddd\u79bb
    \n distance.assign(V, INT_MAX);
    \n distance[s] = 0;<\/p>\n

    \/\/ 2. \u8fdb\u884cV-1\u6b21\u677e\u5f1b\u64cd\u4f5c
    \n for (int i = 1; i <= V – 1; ++i) {
    \n bool updated = false;
    \n for (const Edge& e : edges) {
    \n if (distance[e.u] != INT_MAX && distance[e.v] > distance[e.u] + e.weight) {
    \n distance[e.v] = distance[e.u] + e.weight;
    \n updated = true;
    \n }
    \n }
    \n \/\/ \u5982\u679c\u6ca1\u6709\u66f4\u65b0，\u53ef\u4ee5\u63d0\u524d\u9000\u51fa
    \n if (!updated) break;
    \n }<\/p>\n

    \/\/ 3. \u68c0\u6d4b\u8d1f\u6743\u56de\u8def
    \n for (const Edge& e : edges) {
    \n if (distance[e.u] != INT_MAX && distance[e.v] > distance[e.u] + e.weight) {
    \n return true; \/\/ \u5b58\u5728\u8d1f\u6743\u56de\u8def
    \n }
    \n }
    \n return false; \/\/ \u4e0d\u5b58\u5728\u8d1f\u6743\u56de\u8def
    \n}<\/p>\n

    int main() {
    \n \/\/ \u793a\u4f8b\u56fe: \u5305\u542b5\u4e2a\u8282\u70b9(0-4)
    \n int V = 5;
    \n vector<Edge> edges;<\/p>\n

    \/\/ \u6dfb\u52a0\u8fb9
    \n edges.emplace_back(0, 1, -1);
    \n edges.emplace_back(0, 2, 4);
    \n edges.emplace_back(1, 2, 3);
    \n edges.emplace_back(1, 3, 2);
    \n edges.emplace_back(1, 4, 2);
    \n edges.emplace_back(3, 1, 1);
    \n edges.emplace_back(3, 2, 5);
    \n edges.emplace_back(4, 3, -3);<\/p>\n

    int source = 0;
    \n vector<int> distance;<\/p>\n

    bool hasNegativeCycle = bellmanFord(edges, V, source, distance);<\/p>\n

    if (hasNegativeCycle) {
    \n cout << "\u56fe\u4e2d\u5b58\u5728\u8d1f\u6743\u56de\u8def" << endl;
    \n } else {
    \n cout << "\u4ece\u6e90\u8282\u70b9 " << source << " \u5230\u5404\u8282\u70b9\u7684\u6700\u77ed\u8ddd\u79bb:" << endl;
    \n for (int i = 0; i < V; ++i) {
    \n if (distance[i] == INT_MAX) {
    \n cout << "\u8282\u70b9 " << i << ": \u4e0d\u53ef\u8fbe" << endl;
    \n } else {
    \n cout << "\u8282\u70b9 " << i << ": " << distance[i] << endl;
    \n }
    \n }
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h4>\n

    <\/h4>\n

    \u7efc\u5408\u6848\u4f8b\u53ca\u5e94\u7528<\/span><\/h4>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u4e0b\u9762\u662f\u4e00\u4e2a\u7269\u6d41\u8def\u5f84\u89c4\u5212\u7684\u6848\u4f8b，\u5c55\u793a\u5982\u4f55\u4f7f\u7528 Bellman-Ford \u7b97\u6cd5\u5904\u7406\u53ef\u80fd\u5305\u542b\u8d1f\u6743\u8fb9（\u5982\u6298\u6263\u8def\u7ebf）\u7684\u573a\u666f：<\/p>\n

    #include <iostream>
    \n#include <vector>
    \n#include <climits>
    \n#include <string>
    \nusing namespace std;<\/p>\n

    struct Edge {
    \n int from;
    \n int to;
    \n int cost; \/\/ \u53ef\u4ee5\u662f\u8d1f\u6570，\u8868\u793a\u6298\u6263
    \n Edge(int f, int t, int c) : from(f), to(t), cost(c) {}
    \n};<\/p>\n

    bool bellmanFord(const vector<Edge>& edges, int V, int s, vector<int>& distance, vector<int>& parent) {
    \n distance.assign(V, INT_MAX);
    \n parent.assign(V, -1);
    \n distance[s] = 0;<\/p>\n

    for (int i = 1; i <= V – 1; ++i) {
    \n for (const Edge& e : edges) {
    \n if (distance[e.from] != INT_MAX && distance[e.to] > distance[e.from] + e.cost) {
    \n distance[e.to] = distance[e.from] + e.cost;
    \n parent[e.to] = e.from;
    \n }
    \n }
    \n }<\/p>\n

    for (const Edge& e : edges) {
    \n if (distance[e.from] != INT_MAX && distance[e.to] > distance[e.from] + e.cost) {
    \n return true;
    \n }
    \n }
    \n return false;
    \n}<\/p>\n

    \/\/ \u6253\u5370\u4ece\u6e90\u8282\u70b9\u5230\u76ee\u6807\u8282\u70b9\u7684\u8def\u5f84
    \nvoid printPath(int target, const vector<int>& parent) {
    \n if (parent[target] == -1) {
    \n cout << target;
    \n return;
    \n }
    \n printPath(parent[target], parent);
    \n cout << " -> " << target;
    \n}<\/p>\n

    int main() {
    \n \/\/ \u7269\u6d41\u7f51\u7edc: 5\u4e2a\u4ed3\u5e93\u8282\u70b9(0-4)
    \n int V = 5;
    \n vector<Edge> edges;<\/p>\n

    \/\/ \u6dfb\u52a0\u8fd0\u8f93\u8def\u7ebf\u53ca\u6210\u672c(\u8d1f\u6570\u8868\u793a\u6298\u6263)
    \n edges.emplace_back(0, 1, 10); \/\/ \u4ed3\u5e930\u52301，\u6210\u672c10
    \n edges.emplace_back(0, 2, 30); \/\/ \u4ed3\u5e930\u52302，\u6210\u672c30
    \n edges.emplace_back(0, 3, 100); \/\/ \u4ed3\u5e930\u52303，\u6210\u672c100
    \n edges.emplace_back(1, 2, 5); \/\/ \u4ed3\u5e931\u52302，\u6210\u672c5
    \n edges.emplace_back(2, 3, 20); \/\/ \u4ed3\u5e932\u52303，\u6210\u672c20
    \n edges.emplace_back(3, 4, 10); \/\/ \u4ed3\u5e933\u52304，\u6210\u672c10
    \n edges.emplace_back(2, 4, 60); \/\/ \u4ed3\u5e932\u52304，\u6210\u672c60
    \n edges.emplace_back(1, 4, -50); \/\/ \u4ed3\u5e931\u52304，\u6298\u6263\u8def\u7ebf，\u6210\u672c-50(\u5b9e\u9645\u662f\u6536\u76ca)<\/p>\n

    int source = 0;
    \n vector<int> distance, parent;<\/p>\n

    bool hasNegativeCycle = bellmanFord(edges, V, source, distance, parent);<\/p>\n

    if (hasNegativeCycle) {
    \n cout << "\u7269\u6d41\u7f51\u7edc\u4e2d\u5b58\u5728\u5f02\u5e38\u6298\u6263\u5faa\u73af，\u53ef\u80fd\u5bfc\u81f4\u6210\u672c\u65e0\u9650\u964d\u4f4e！" << endl;
    \n } else {
    \n cout << "\u4ece\u4ed3\u5e93 " << source << " \u5230\u5404\u4ed3\u5e93\u7684\u6700\u4f4e\u6210\u672c:" << endl;
    \n for (int i = 0; i < V; ++i) {
    \n if (distance[i] == INT_MAX) {
    \n cout << "\u4ed3\u5e93 " << i << ": \u4e0d\u53ef\u8fbe" << endl;
    \n } else {
    \n cout << "\u4ed3\u5e93 " << i << ": \u6210\u672c " << distance[i] << ", \u8def\u5f84: ";
    \n printPath(i, parent);
    \n cout << endl;
    \n }
    \n }
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h3>\n

    <\/h4>\n

    24.2 \u6709\u5411\u65e0\u73af\u56fe\u4e2d\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898<\/span><\/h3>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u6709\u5411\u65e0\u73af\u56fe（DAG）\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u62d3\u6251\u6392\u5e8f\u7ed3\u5408\u677e\u5f1b\u64cd\u4f5c\u9ad8\u6548\u89e3\u51b3<\/span>，\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O (V+E)，\u4f18\u4e8e Bellman-Ford \u7b97\u6cd5\u3002<\/p>\n

    \u7b97\u6cd5\u601d\u60f3<\/span><\/h4>\n
  • \u5bf9 DAG \u8fdb\u884c\u62d3\u6251\u6392\u5e8f，\u5f97\u5230\u4e00\u4e2a\u7ebf\u6027\u6392\u5217\u7684\u8282\u70b9\u5e8f\u5217\u3002<\/li>\n
  • \u521d\u59cb\u5316\u8ddd\u79bb\u6570\u7ec4：\u6e90\u8282\u70b9\u8ddd\u79bb\u4e3a 0，\u5176\u4ed6\u8282\u70b9\u4e3a\u65e0\u7a77\u5927\u3002<\/li>\n
  • \u6309\u7167\u62d3\u6251\u6392\u5e8f\u7684\u987a\u5e8f，\u5bf9\u6bcf\u4e2a\u8282\u70b9\u7684\u6240\u6709\u51fa\u8fb9\u8fdb\u884c\u677e\u5f1b\u64cd\u4f5c\u3002<\/li>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7531\u4e8e DAG \u4e2d\u6ca1\u6709\u73af，\u4e14\u6211\u4eec\u6309\u7167\u62d3\u6251\u987a\u5e8f\u5904\u7406\u8282\u70b9，\u6bcf\u4e2a\u8282\u70b9\u53ea\u4f1a\u88ab\u5904\u7406\u4e00\u6b21，\u56e0\u6b64\u7b97\u6cd5\u6548\u7387\u5f88\u9ad8\u3002\u8be5\u7b97\u6cd5\u53ef\u4ee5\u5904\u7406\u8d1f\u6743\u8fb9，\u4f46\u4e0d\u80fd\u5904\u7406\u6709\u73af\u7684\u56fe\u3002<\/p>\n

    <\/h4>\n

    \u4ee3\u7801\u5b9e\u73b0<\/span><\/h4>\n

    #include <iostream>
    \n#include <vector>
    \n#include <stack>
    \n#include <climits>
    \nusing namespace std;<\/p>\n

    \/**
    \n * \u5bf9DAG\u8fdb\u884c\u62d3\u6251\u6392\u5e8f
    \n * @param adj \u90bb\u63a5\u8868\u8868\u793a\u7684\u56fe
    \n * @param V \u8282\u70b9\u6570\u91cf
    \n * @return \u62d3\u6251\u6392\u5e8f\u7ed3\u679c
    \n *\/
    \nvector<int> topologicalSort(const vector<vector<pair<int, int>>>& adj, int V) {
    \n vector<int> inDegree(V, 0);
    \n \/\/ \u8ba1\u7b97\u5165\u5ea6
    \n for (int u = 0; u < V; ++u) {
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n inDegree[v]++;
    \n }
    \n }<\/p>\n

    \/\/ \u4f7f\u7528\u6808\u5b58\u50a8\u5165\u5ea6\u4e3a0\u7684\u8282\u70b9
    \n stack<int> s;
    \n for (int u = 0; u < V; ++u) {
    \n if (inDegree[u] == 0) {
    \n s.push(u);
    \n }
    \n }<\/p>\n

    vector<int> topoOrder;
    \n while (!s.empty()) {
    \n int u = s.top();
    \n s.pop();
    \n topoOrder.push_back(u);<\/p>\n

    \/\/ \u51cf\u5c11\u76f8\u90bb\u8282\u70b9\u7684\u5165\u5ea6
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n inDegree[v]–;
    \n if (inDegree[v] == 0) {
    \n s.push(v);
    \n }
    \n }
    \n }<\/p>\n

    return topoOrder;
    \n}<\/p>\n

    \/**
    \n * DAG\u4e2d\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u7b97\u6cd5
    \n * @param adj \u90bb\u63a5\u8868\u8868\u793a\u7684\u56fe
    \n * @param V \u8282\u70b9\u6570\u91cf
    \n * @param s \u6e90\u8282\u70b9
    \n * @param distance \u5b58\u50a8\u6700\u77ed\u8ddd\u79bb\u7684\u6570\u7ec4
    \n * @param parent \u5b58\u50a8\u524d\u9a71\u8282\u70b9\u7684\u6570\u7ec4
    \n *\/
    \nvoid dagShortestPath(const vector<vector<pair<int, int>>>& adj, int V, int s,
    \n vector<int>& distance, vector<int>& parent) {
    \n \/\/ 1. \u62d3\u6251\u6392\u5e8f
    \n vector<int> topoOrder = topologicalSort(adj, V);<\/p>\n

    \/\/ 2. \u521d\u59cb\u5316
    \n distance.assign(V, INT_MAX);
    \n parent.assign(V, -1);
    \n distance[s] = 0;<\/p>\n

    \/\/ 3. \u6309\u62d3\u6251\u987a\u5e8f\u677e\u5f1b\u8fb9
    \n for (int u : topoOrder) {
    \n if (distance[u] != INT_MAX) { \/\/ \u4ec5\u5904\u7406\u53ef\u8fbe\u8282\u70b9
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n int weight = edge.second;
    \n \/\/ \u677e\u5f1b\u64cd\u4f5c
    \n if (distance[v] > distance[u] + weight) {
    \n distance[v] = distance[u] + weight;
    \n parent[v] = u;
    \n }
    \n }
    \n }
    \n }
    \n}<\/p>\n

    \/\/ \u6253\u5370\u8def\u5f84
    \nvoid printPath(int v, const vector<int>& parent) {
    \n if (parent[v] == -1) {
    \n cout << v;
    \n return;
    \n }
    \n printPath(parent[v], parent);
    \n cout << " -> " << v;
    \n}<\/p>\n

    int main() {
    \n \/\/ \u793a\u4f8bDAG: 5\u4e2a\u8282\u70b9(0-4)
    \n int V = 5;
    \n vector<vector<pair<int, int>>> adj(V);<\/p>\n

    \/\/ \u6dfb\u52a0\u8fb9 (u, v, weight)
    \n adj[0].emplace_back(1, 5);
    \n adj[0].emplace_back(2, 3);
    \n adj[1].emplace_back(2, 2);
    \n adj[1].emplace_back(3, 6);
    \n adj[2].emplace_back(3, 7);
    \n adj[2].emplace_back(4, 4);
    \n adj[2].emplace_back(1, 1);
    \n adj[3].emplace_back(4, -1);
    \n adj[4].emplace_back(3, -2);<\/p>\n

    int source = 1;
    \n vector<int> distance, parent;<\/p>\n

    dagShortestPath(adj, V, source, distance, parent);<\/p>\n

    cout << "\u4ece\u6e90\u8282\u70b9 " << source << " \u5230\u5404\u8282\u70b9\u7684\u6700\u77ed\u8ddd\u79bb:" << endl;
    \n for (int i = 0; i < V; ++i) {
    \n if (distance[i] == INT_MAX) {
    \n cout << "\u8282\u70b9 " << i << ": \u4e0d\u53ef\u8fbe" << endl;
    \n } else {
    \n cout << "\u8282\u70b9 " << i << ": " << distance[i] << ", \u8def\u5f84: ";
    \n printPath(i, parent);
    \n cout << endl;
    \n }
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h4>\n

    <\/h4>\n

    \u7efc\u5408\u6848\u4f8b\u53ca\u5e94\u7528<\/span><\/h4>\n

    \u4e0b\u9762\u662f\u4e00\u4e2a\u9879\u76ee\u8c03\u5ea6\u7684\u6848\u4f8b，\u5c55\u793a\u5982\u4f55\u4f7f\u7528 DAG \u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u89e3\u51b3\u5173\u952e\u8def\u5f84\u95ee\u9898：<\/p>\n

    #include <iostream>
    \n#include <vector>
    \n#include <stack>
    \n#include <climits>
    \n#include <algorithm>
    \nusing namespace std;<\/p>\n

    \/\/ \u5bf9DAG\u8fdb\u884c\u62d3\u6251\u6392\u5e8f
    \nvector<int> topologicalSort(const vector<vector<pair<int, int>>>& adj, int V) {
    \n vector<int> inDegree(V, 0);
    \n for (int u = 0; u < V; ++u) {
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n inDegree[v]++;
    \n }
    \n }<\/p>\n

    stack<int> s;
    \n for (int u = 0; u < V; ++u) {
    \n if (inDegree[u] == 0) {
    \n s.push(u);
    \n }
    \n }<\/p>\n

    vector<int> topoOrder;
    \n while (!s.empty()) {
    \n int u = s.top();
    \n s.pop();
    \n topoOrder.push_back(u);<\/p>\n

    for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n inDegree[v]–;
    \n if (inDegree[v] == 0) {
    \n s.push(v);
    \n }
    \n }
    \n }<\/p>\n

    return topoOrder;
    \n}<\/p>\n

    \/\/ \u8ba1\u7b97\u6700\u957f\u8def\u5f84（\u5173\u952e\u8def\u5f84）
    \nvoid criticalPathAnalysis(const vector<vector<pair<int, int>>>& adj, int V,
    \n vector<int>& earliest, vector<int>& latest, vector<int>& parent) {
    \n vector<int> topoOrder = topologicalSort(adj, V);<\/p>\n

    \/\/ \u8ba1\u7b97\u6700\u65e9\u5f00\u59cb\u65f6\u95f4（\u6700\u957f\u8def\u5f84）
    \n earliest.assign(V, 0);
    \n parent.assign(V, -1);<\/p>\n

    for (int u : topoOrder) {
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n int weight = edge.second;
    \n if (earliest[v] < earliest[u] + weight) {
    \n earliest[v] = earliest[u] + weight;
    \n parent[v] = u;
    \n }
    \n }
    \n }<\/p>\n

    \/\/ \u8ba1\u7b97\u6700\u665a\u5f00\u59cb\u65f6\u95f4
    \n latest.assign(V, earliest.back()); \/\/ \u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u6700\u665a\u65f6\u95f4\u7b49\u4e8e\u6700\u65e9\u65f6\u95f4<\/p>\n

    \/\/ \u53cd\u8f6c\u62d3\u6251\u6392\u5e8f
    \n reverse(topoOrder.begin(), topoOrder.end());<\/p>\n

    for (int u : topoOrder) {
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n int weight = edge.second;
    \n if (latest[u] > latest[v] – weight) {
    \n latest[u] = latest[v] – weight;
    \n }
    \n }
    \n }
    \n}<\/p>\n

    \/\/ \u6253\u5370\u5173\u952e\u8def\u5f84
    \nvoid printCriticalPath(int v, const vector<int>& parent) {
    \n if (parent[v] == -1) {
    \n cout << v;
    \n return;
    \n }
    \n printCriticalPath(parent[v], parent);
    \n cout << " -> " << v;
    \n}<\/p>\n

    int main() {
    \n \/\/ \u9879\u76ee\u4efb\u52a1\u8282\u70b9: 0(\u5f00\u59cb), 1-6(\u4efb\u52a1), 7(\u7ed3\u675f)
    \n int V = 8;
    \n vector<vector<pair<int, int>>> adj(V);<\/p>\n

    \/\/ \u6dfb\u52a0\u4efb\u52a1\u4f9d\u8d56\u548c\u6301\u7eed\u65f6\u95f4 (u, v, \u6301\u7eed\u65f6\u95f4)
    \n adj[0].emplace_back(1, 3); \/\/ \u5f00\u59cb -> \u4efb\u52a11, 3\u5929
    \n adj[0].emplace_back(2, 2); \/\/ \u5f00\u59cb -> \u4efb\u52a12, 2\u5929
    \n adj[1].emplace_back(3, 4); \/\/ \u4efb\u52a11 -> \u4efb\u52a13, 4\u5929
    \n adj[1].emplace_back(4, 3); \/\/ \u4efb\u52a11 -> \u4efb\u52a14, 3\u5929
    \n adj[2].emplace_back(4, 2); \/\/ \u4efb\u52a12 -> \u4efb\u52a14, 2\u5929
    \n adj[3].emplace_back(5, 1); \/\/ \u4efb\u52a13 -> \u4efb\u52a15, 1\u5929
    \n adj[4].emplace_back(5, 2); \/\/ \u4efb\u52a14 -> \u4efb\u52a15, 2\u5929
    \n adj[4].emplace_back(6, 3); \/\/ \u4efb\u52a14 -> \u4efb\u52a16, 3\u5929
    \n adj[5].emplace_back(7, 3); \/\/ \u4efb\u52a15 -> \u7ed3\u675f, 3\u5929
    \n adj[6].emplace_back(7, 2); \/\/ \u4efb\u52a16 -> \u7ed3\u675f, 2\u5929<\/p>\n

    vector<int> earliest, latest, parent;
    \n criticalPathAnalysis(adj, V, earliest, latest, parent);<\/p>\n

    cout << "\u9879\u76ee\u8ba1\u5212\u5206\u6790\u7ed3\u679c:" << endl;
    \n cout << "\u9879\u76ee\u6700\u77ed\u5b8c\u6210\u65f6\u95f4: " << earliest.back() << "\u5929" << endl;
    \n cout << endl;<\/p>\n

    cout << "\u5404\u4efb\u52a1\u65f6\u95f4\u5206\u6790:" << endl;
    \n cout << "\u4efb\u52a1ID\\\\t\u6700\u65e9\u5f00\u59cb\\\\t\u6700\u665a\u5f00\u59cb\\\\t\u6d6e\u52a8\u65f6\u95f4" << endl;
    \n for (int i = 1; i < V – 1; ++i) { \/\/ \u8df3\u8fc7\u5f00\u59cb(0)\u548c\u7ed3\u675f(V-1)\u8282\u70b9
    \n int slack = latest[i] – earliest[i];
    \n cout << i << "\\\\t" << earliest[i] << "\\\\t\\\\t" << latest[i] << "\\\\t\\\\t" << slack << endl;
    \n }<\/p>\n

    cout << endl << "\u5173\u952e\u8def\u5f84(\u65e0\u6d6e\u52a8\u65f6\u95f4\u7684\u4efb\u52a1\u5e8f\u5217): ";
    \n printCriticalPath(V – 1, parent);
    \n cout << endl;<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h3>\n

    <\/h4>\n

    24.3 Dijkstra \u7b97\u6cd5<\/span><\/h3>\n

    \"\"<\/p>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Dijkstra \u7b97\u6cd5\u662f\u4e00\u79cd\u8d2a\u5fc3\u7b97\u6cd5，\u7528\u4e8e\u6c42\u89e3\u975e\u8d1f\u6743\u56fe\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898<\/span>\u3002\u5b83\u7684\u6548\u7387\u9ad8\u4e8e Bellman-Ford \u7b97\u6cd5，\u5c24\u5176\u5728\u4f7f\u7528\u4f18\u5148\u961f\u5217<\/span>\u4f18\u5316\u540e，\u6027\u80fd\u8868\u73b0\u4f18\u5f02\u3002<\/p>\n

    \u7b97\u6cd5\u601d\u60f3<\/span><\/h4>\n
  • \u521d\u59cb\u5316：\u6e90\u8282\u70b9\u8ddd\u79bb\u4e3a 0，\u5176\u4ed6\u8282\u70b9\u8ddd\u79bb\u4e3a\u65e0\u7a77\u5927\u3002\u7ef4\u62a4\u4e00\u4e2a\u4f18\u5148\u961f\u5217（\u6700\u5c0f\u5806），\u7528\u4e8e\u9009\u62e9\u5f53\u524d\u8ddd\u79bb\u6700\u5c0f\u7684\u8282\u70b9\u3002<\/li>\n
  • \u9009\u62e9\u8282\u70b9：\u6bcf\u6b21\u4ece\u4f18\u5148\u961f\u5217\u4e2d\u53d6\u51fa\u8ddd\u79bb\u6700\u5c0f\u7684\u8282\u70b9 u\u3002<\/li>\n
  • \u677e\u5f1b\u64cd\u4f5c：\u5bf9\u8282\u70b9 u \u7684\u6240\u6709\u51fa\u8fb9\u8fdb\u884c\u677e\u5f1b\u64cd\u4f5c\u3002\u5982\u679c\u901a\u8fc7 u \u5230\u8fbe v \u7684\u8def\u5f84\u6bd4\u5f53\u524d\u5df2\u77e5\u8def\u5f84\u66f4\u77ed，\u5219\u66f4\u65b0 v \u7684\u8ddd\u79bb，\u5e76\u5c06 v \u52a0\u5165\u4f18\u5148\u961f\u5217\u3002<\/li>\n
  • \u6807\u8bb0\u5df2\u5904\u7406：\u4e00\u65e6\u8282\u70b9 u \u88ab\u53d6\u51fa\u5e76\u5904\u7406，\u5c31\u6807\u8bb0\u4e3a\u5df2\u5904\u7406，\u4e0d\u518d\u91cd\u65b0\u5904\u7406\u3002<\/li>\n

    \"\"<\/p>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7531\u4e8e\u56fe\u4e2d\u6240\u6709\u8fb9\u7684\u6743\u91cd\u90fd\u662f\u975e\u8d1f\u6570，\u4e00\u65e6\u67d0\u4e2a\u8282\u70b9\u88ab\u6807\u8bb0\u4e3a\u5df2\u5904\u7406，\u6211\u4eec\u5c31\u627e\u5230\u4e86\u4ece\u6e90\u8282\u70b9\u5230\u8be5\u8282\u70b9\u7684\u6700\u77ed\u8def\u5f84，\u65e0\u9700\u518d\u6b21\u8003\u8651\u3002<\/p>\n

    <\/h4>\n

    \u4ee3\u7801\u5b9e\u73b0<\/span><\/h4>\n

    #include <iostream>
    \n#include <vector>
    \n#include <queue>
    \n#include <climits>
    \n#include <utility>
    \nusing namespace std;<\/p>\n

    \/**
    \n * Dijkstra\u7b97\u6cd5\u5b9e\u73b0(\u4f7f\u7528\u4f18\u5148\u961f\u5217\u4f18\u5316)
    \n * @param adj \u90bb\u63a5\u8868\u8868\u793a\u7684\u56fe
    \n * @param V \u8282\u70b9\u6570\u91cf
    \n * @param s \u6e90\u8282\u70b9
    \n * @param distance \u5b58\u50a8\u6700\u77ed\u8ddd\u79bb\u7684\u6570\u7ec4
    \n * @param parent \u5b58\u50a8\u524d\u9a71\u8282\u70b9\u7684\u6570\u7ec4
    \n *\/
    \nvoid dijkstra(const vector<vector<pair<int, int>>>& adj, int V, int s,
    \n vector<int>& distance, vector<int>& parent) {
    \n \/\/ \u521d\u59cb\u5316
    \n distance.assign(V, INT_MAX);
    \n parent.assign(V, -1);
    \n distance[s] = 0;<\/p>\n

    \/\/ \u4f18\u5148\u961f\u5217: (\u8ddd\u79bb, \u8282\u70b9), \u4f7f\u7528\u6700\u5c0f\u5806
    \n priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq;
    \n pq.push({0, s});<\/p>\n

    \/\/ \u8bb0\u5f55\u5df2\u5904\u7406\u7684\u8282\u70b9
    \n vector<bool> processed(V, false);<\/p>\n

    while (!pq.empty()) {
    \n \/\/ \u63d0\u53d6\u8ddd\u79bb\u6700\u5c0f\u7684\u8282\u70b9
    \n int u = pq.top().second;
    \n pq.pop();<\/p>\n

    \/\/ \u5982\u679c\u5df2\u5904\u7406，\u8df3\u8fc7
    \n if (processed[u]) continue;
    \n processed[u] = true;<\/p>\n

    \/\/ \u677e\u5f1b\u6240\u6709\u51fa\u8fb9
    \n for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n int weight = edge.second;<\/p>\n

    \/\/ \u677e\u5f1b\u64cd\u4f5c
    \n if (distance[v] > distance[u] + weight) {
    \n distance[v] = distance[u] + weight;
    \n parent[v] = u;
    \n pq.push({distance[v], v});
    \n }
    \n }
    \n }
    \n}<\/p>\n

    \/\/ \u6253\u5370\u8def\u5f84
    \nvoid printPath(int v, const vector<int>& parent) {
    \n if (parent[v] == -1) {
    \n cout << v;
    \n return;
    \n }
    \n printPath(parent[v], parent);
    \n cout << " -> " << v;
    \n}<\/p>\n

    int main() {
    \n \/\/ \u793a\u4f8b\u56fe: 6\u4e2a\u8282\u70b9(0-5)
    \n int V = 6;
    \n vector<vector<pair<int, int>>> adj(V);<\/p>\n

    \/\/ \u6dfb\u52a0\u8fb9 (u, v, weight)
    \n adj[0].emplace_back(1, 7);
    \n adj[0].emplace_back(2, 9);
    \n adj[0].emplace_back(5, 14);
    \n adj[1].emplace_back(0, 7);
    \n adj[1].emplace_back(2, 10);
    \n adj[1].emplace_back(3, 15);
    \n adj[2].emplace_back(0, 9);
    \n adj[2].emplace_back(1, 10);
    \n adj[2].emplace_back(3, 11);
    \n adj[2].emplace_back(5, 2);
    \n adj[3].emplace_back(1, 15);
    \n adj[3].emplace_back(2, 11);
    \n adj[3].emplace_back(4, 6);
    \n adj[4].emplace_back(3, 6);
    \n adj[4].emplace_back(5, 9);
    \n adj[5].emplace_back(0, 14);
    \n adj[5].emplace_back(2, 2);
    \n adj[5].emplace_back(4, 9);<\/p>\n

    int source = 0;
    \n vector<int> distance, parent;<\/p>\n

    dijkstra(adj, V, source, distance, parent);<\/p>\n

    cout << "\u4ece\u6e90\u8282\u70b9 " << source << " \u5230\u5404\u8282\u70b9\u7684\u6700\u77ed\u8ddd\u79bb:" << endl;
    \n for (int i = 0; i < V; ++i) {
    \n if (distance[i] == INT_MAX) {
    \n cout << "\u8282\u70b9 " << i << ": \u4e0d\u53ef\u8fbe" << endl;
    \n } else {
    \n cout << "\u8282\u70b9 " << i << ": " << distance[i] << ", \u8def\u5f84: ";
    \n printPath(i, parent);
    \n cout << endl;
    \n }
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h4>\n

    <\/h4>\n

    \u7efc\u5408\u6848\u4f8b\u53ca\u5e94\u7528<\/span><\/h4>\n

    \u4e0b\u9762\u662f\u4e00\u4e2a\u57ce\u5e02\u4ea4\u901a\u5bfc\u822a\u7cfb\u7edf\u7684\u6848\u4f8b，\u5c55\u793a\u5982\u4f55\u4f7f\u7528 Dijkstra \u7b97\u6cd5\u89c4\u5212\u6700\u77ed\u8def\u5f84：<\/p>\n

    #include <iostream>
    \n#include <vector>
    \n#include <queue>
    \n#include <climits>
    \n#include <utility>
    \n#include <string>
    \nusing namespace std;<\/p>\n

    \/\/ \u57ce\u5e02\u8282\u70b9\u7c7b
    \nclass CityGraph {
    \nprivate:
    \n vector<vector<pair<int, int>>> adj; \/\/ \u90bb\u63a5\u8868: (\u57ce\u5e02ID, \u8ddd\u79bb)
    \n vector<string> cityNames; \/\/ \u57ce\u5e02\u540d\u79f0<\/p>\n

    public:
    \n \/\/ \u6784\u9020\u51fd\u6570
    \n CityGraph(int n) : adj(n) {}<\/p>\n

    \/\/ \u6dfb\u52a0\u57ce\u5e02\u540d\u79f0
    \n void addCityName(int id, const string& name) {
    \n if (id >= cityNames.size()) {
    \n cityNames.resize(id + 1);
    \n }
    \n cityNames[id] = name;
    \n }<\/p>\n

    \/\/ \u6dfb\u52a0\u9053\u8def
    \n void addRoad(int from, int to, int distance) {
    \n adj[from].emplace_back(to, distance);
    \n adj[to].emplace_back(from, distance); \/\/ \u5047\u8bbe\u9053\u8def\u662f\u53cc\u5411\u7684
    \n }<\/p>\n

    \/\/ \u67e5\u627e\u6700\u77ed\u8def\u5f84
    \n void findShortestPath(int start, int end, vector<int>& distance, vector<int>& parent) {
    \n int V = adj.size();
    \n distance.assign(V, INT_MAX);
    \n parent.assign(V, -1);
    \n distance[start] = 0;<\/p>\n

    priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq;
    \n pq.push({0, start});<\/p>\n

    vector<bool> processed(V, false);<\/p>\n

    while (!pq.empty()) {
    \n int u = pq.top().second;
    \n pq.pop();<\/p>\n

    if (u == end) break; \/\/ \u63d0\u524d\u7ed3\u675f，\u627e\u5230\u76ee\u6807\u57ce\u5e02
    \n if (processed[u]) continue;
    \n processed[u] = true;<\/p>\n

    for (const auto& edge : adj[u]) {
    \n int v = edge.first;
    \n int dist = edge.second;<\/p>\n

    if (distance[v] > distance[u] + dist) {
    \n distance[v] = distance[u] + dist;
    \n parent[v] = u;
    \n pq.push({distance[v], v});
    \n }
    \n }
    \n }
    \n }<\/p>\n

    \/\/ \u6253\u5370\u8def\u5f84
    \n void printPath(int v, const vector<int>& parent) {
    \n if (parent[v] == -1) {
    \n cout << cityNames[v];
    \n return;
    \n }
    \n printPath(parent[v], parent);
    \n cout << " -> " << cityNames[v];
    \n }<\/p>\n

    \/\/ \u83b7\u53d6\u57ce\u5e02\u540d\u79f0
    \n string getCityName(int id) const {
    \n if (id >= 0 && id < cityNames.size()) {
    \n return cityNames[id];
    \n }
    \n return "\u672a\u77e5\u57ce\u5e02";
    \n }
    \n};<\/p>\n

    int main() {
    \n \/\/ \u521b\u5efa\u57ce\u5e02\u56fe(8\u4e2a\u57ce\u5e02)
    \n CityGraph graph(8);<\/p>\n

    \/\/ \u6dfb\u52a0\u57ce\u5e02\u540d\u79f0
    \n graph.addCityName(0, "\u5317\u4eac");
    \n graph.addCityName(1, "\u5929\u6d25");
    \n graph.addCityName(2, "\u77f3\u5bb6\u5e84");
    \n graph.addCityName(3, "\u592a\u539f");
    \n graph.addCityName(4, "\u6d4e\u5357");
    \n graph.addCityName(5, "\u90d1\u5dde");
    \n graph.addCityName(6, "\u897f\u5b89");
    \n graph.addCityName(7, "\u6b66\u6c49");<\/p>\n

    \/\/ \u6dfb\u52a0\u57ce\u5e02\u95f4\u9053\u8def(\u8ddd\u79bb\u5355\u4f4d: \u516c\u91cc)
    \n graph.addRoad(0, 1, 137); \/\/ \u5317\u4eac-\u5929\u6d25
    \n graph.addRoad(0, 2, 283); \/\/ \u5317\u4eac-\u77f3\u5bb6\u5e84
    \n graph.addRoad(0, 4, 497); \/\/ \u5317\u4eac-\u6d4e\u5357
    \n graph.addRoad(1, 4, 357); \/\/ \u5929\u6d25-\u6d4e\u5357
    \n graph.addRoad(2, 3, 231); \/\/ \u77f3\u5bb6\u5e84-\u592a\u539f
    \n graph.addRoad(2, 5, 412); \/\/ \u77f3\u5bb6\u5e84-\u90d1\u5dde
    \n graph.addRoad(3, 6, 511); \/\/ \u592a\u539f-\u897f\u5b89
    \n graph.addRoad(4, 5, 374); \/\/ \u6d4e\u5357-\u90d1\u5dde
    \n graph.addRoad(5, 6, 480); \/\/ \u90d1\u5dde-\u897f\u5b89
    \n graph.addRoad(5, 7, 536); \/\/ \u90d1\u5dde-\u6b66\u6c49
    \n graph.addRoad(6, 7, 691); \/\/ \u897f\u5b89-\u6b66\u6c49<\/p>\n

    int start = 0; \/\/ \u5317\u4eac
    \n int end = 7; \/\/ \u6b66\u6c49
    \n vector<int> distance, parent;<\/p>\n

    graph.findShortestPath(start, end, distance, parent);<\/p>\n

    cout << "\u57ce\u5e02\u5bfc\u822a\u7ed3\u679c:" << endl;
    \n cout << "\u4ece " << graph.getCityName(start) << " \u5230 " << graph.getCityName(end)
    \n << " \u7684\u6700\u77ed\u8ddd\u79bb\u4e3a: " << distance[end] << " \u516c\u91cc" << endl;
    \n cout << "\u63a8\u8350\u8def\u7ebf: ";
    \n graph.printPath(end, parent);
    \n cout << endl;<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h3>\n

    <\/h4>\n

    24.4 \u5dee\u5206\u7ea6\u675f\u548c\u6700\u77ed\u8def\u5f84<\/span><\/h3>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u662f\u4e00\u7c7b\u7279\u6b8a\u7684\u7ebf\u6027\u89c4\u5212\u95ee\u9898，\u5b83\u53ef\u4ee5\u8f6c\u5316\u4e3a\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u6765\u6c42\u89e3\u3002\u901a\u8fc7\u5c06\u6bcf\u4e2a\u7ea6\u675f\u6761\u4ef6\u8f6c\u5316\u4e3a\u56fe\u4e2d\u7684\u4e00\u6761\u8fb9，\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u7684\u89e3\u3002<\/p>\n

    \u7b97\u6cd5\u601d\u60f3<\/span><\/h4>\n
  • \u5efa\u6a21：\u5c06\u6bcf\u4e2a\u53d8\u91cf(x_i)\u8868\u793a\u4e3a\u56fe\u4e2d\u7684\u4e00\u4e2a\u8282\u70b9i\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u7ea6\u675f\u6761\u4ef6(x_j – x_i \\\\leq w)，\u521b\u5efa\u4e00\u6761\u4ece\u8282\u70b9i\u5230\u8282\u70b9j\u3001\u6743\u91cd\u4e3aw\u7684\u8fb9\u3002<\/li>\n
  • \u5f15\u5165\u8d85\u7ea7\u6e90\u8282\u70b9：\u4e3a\u4e86\u4fdd\u8bc1\u56fe\u7684\u8fde\u901a\u6027，\u5f15\u5165\u4e00\u4e2a\u8d85\u7ea7\u6e90\u8282\u70b9 0，\u5e76\u6dfb\u52a0\u4ece 0 \u5230\u6240\u6709\u5176\u4ed6\u8282\u70b9\u7684\u8fb9，\u6743\u91cd\u4e3a 0\u3002<\/li>\n
  • \u6c42\u89e3\u6700\u77ed\u8def\u5f84：\u4f7f\u7528 Bellman-Ford \u7b97\u6cd5\u8ba1\u7b97\u4ece\u8d85\u7ea7\u6e90\u8282\u70b9\u5230\u6240\u6709\u5176\u4ed6\u8282\u70b9\u7684\u6700\u77ed\u8def\u5f84\u3002\u5982\u679c\u56fe\u4e2d\u5b58\u5728\u8d1f\u6743\u56de\u8def，\u5219\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u65e0\u89e3；\u5426\u5219，\u6700\u77ed\u8def\u5f84\u8ddd\u79bb\u5c31\u662f\u7cfb\u7edf\u7684\u4e00\u4e2a\u53ef\u884c\u89e3\u3002<\/li>\n

    \u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u5efa\u6a21\u793a\u610f\u56fe<\/span><\/h4>\n

    \u7ea6\u675f\u6761\u4ef6:
    \nx1 – x0 \u2264 3
    \nx2 – x0 \u2264 5
    \nx3 – x1 \u2264 2
    \nx2 – x1 \u2264 2
    \nx3 – x2 \u2264 -1<\/p>\n

    \u8f6c\u5316\u4e3a\u56fe:
    \n0 -> 1 (\u6743\u91cd3)
    \n0 -> 2 (\u6743\u91cd5)
    \n1 -> 3 (\u6743\u91cd2)
    \n1 -> 2 (\u6743\u91cd2)
    \n2 -> 3 (\u6743\u91cd-1)<\/p>\n

    \u4ee3\u7801\u5b9e\u73b0<\/span><\/h4>\n

    #include <iostream>
    \n#include <vector>
    \n#include <climits>
    \nusing namespace std;<\/p>\n

    struct Edge {
    \n int u, v, weight;
    \n Edge(int u_, int v_, int w_) : u(u_), v(v_), weight(w_) {}
    \n};<\/p>\n

    \/**
    \n * \u6c42\u89e3\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf
    \n * @param edges \u7ea6\u675f\u6761\u4ef6\u8f6c\u5316\u7684\u8fb9
    \n * @param n \u53d8\u91cf\u6570\u91cf
    \n * @param x \u5b58\u50a8\u89e3\u7684\u6570\u7ec4
    \n * @return \u662f\u5426\u6709\u89e3
    \n *\/
    \nbool solveDifferenceConstraints(const vector<Edge>& edges, int n, vector<int>& x) {
    \n \/\/ \u6dfb\u52a0\u8d85\u7ea7\u6e90\u8282\u70b90，\u8fde\u63a5\u5230\u6240\u6709\u5176\u4ed6\u8282\u70b9
    \n vector<Edge> newEdges = edges;
    \n for (int i = 1; i <= n; ++i) {
    \n newEdges.emplace_back(0, i, 0);
    \n }<\/p>\n

    \/\/ \u4f7f\u7528Bellman-Ford\u7b97\u6cd5\u6c42\u89e3
    \n x.assign(n + 1, INT_MAX); \/\/ x[0]\u5230x[n]
    \n x[0] = 0;<\/p>\n

    \/\/ \u677e\u5f1bn\u6b21
    \n for (int i = 0; i < n; ++i) {
    \n for (const Edge& e : newEdges) {
    \n if (x[e.u] != INT_MAX && x[e.v] > x[e.u] + e.weight) {
    \n x[e.v] = x[e.u] + e.weight;
    \n }
    \n }
    \n }<\/p>\n

    \/\/ \u68c0\u6d4b\u8d1f\u6743\u56de\u8def
    \n for (const Edge& e : newEdges) {
    \n if (x[e.u] != INT_MAX && x[e.v] > x[e.u] + e.weight) {
    \n return false; \/\/ \u5b58\u5728\u8d1f\u6743\u56de\u8def，\u65e0\u89e3
    \n }
    \n }<\/p>\n

    return true; \/\/ \u6709\u89e3
    \n}<\/p>\n

    int main() {
    \n \/\/ \u793a\u4f8b: \u6c42\u89e3\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf
    \n \/\/ \u53d8\u91cf: x1, x2, x3
    \n \/\/ \u7ea6\u675f:
    \n \/\/ x1 – x2 <= 1
    \n \/\/ x1 – x3 <= 3
    \n \/\/ x2 – x3 <= 2
    \n \/\/ x3 – x2 <= -1
    \n \/\/ x2 – x1 <= -2<\/p>\n

    int n = 3; \/\/ \u53d8\u91cf\u6570\u91cf
    \n vector<Edge> edges;<\/p>\n

    \/\/ \u6dfb\u52a0\u7ea6\u675f\u5bf9\u5e94\u7684\u8fb9
    \n edges.emplace_back(2, 1, 1); \/\/ x1 – x2 <= 1
    \n edges.emplace_back(3, 1, 3); \/\/ x1 – x3 <= 3
    \n edges.emplace_back(3, 2, 2); \/\/ x2 – x3 <= 2
    \n edges.emplace_back(2, 3, -1); \/\/ x3 – x2 <= -1
    \n edges.emplace_back(1, 2, -2); \/\/ x2 – x1 <= -2<\/p>\n

    vector<int> x;
    \n bool hasSolution = solveDifferenceConstraints(edges, n, x);<\/p>\n

    if (hasSolution) {
    \n cout << "\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u7684\u4e00\u4e2a\u89e3\u4e3a:" << endl;
    \n for (int i = 1; i <= n; ++i) {
    \n cout << "x" << i << " = " << x[i] << endl;
    \n }
    \n } else {
    \n cout << "\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u65e0\u89e3" << endl;
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h4>\n

    <\/h4>\n

    \u7efc\u5408\u6848\u4f8b\u53ca\u5e94\u7528<\/span><\/h4>\n

    \u4e0b\u9762\u662f\u4e00\u4e2a\u8bfe\u7a0b\u5b89\u6392\u7684\u6848\u4f8b，\u5c55\u793a\u5982\u4f55\u4f7f\u7528\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u89e3\u51b3\u5b9e\u9645\u95ee\u9898：<\/p>\n

    #include <iostream>
    \n#include <vector>
    \n#include <climits>
    \n#include <string>
    \nusing namespace std;<\/p>\n

    struct Edge {
    \n int u, v, weight;
    \n Edge(int u_, int v_, int w_) : u(u_), v(v_), weight(w_) {}
    \n};<\/p>\n

    bool solveCourseScheduling(const vector<Edge>& edges, int n, vector<int>& schedule) {
    \n vector<Edge> newEdges = edges;
    \n \/\/ \u6dfb\u52a0\u8d85\u7ea7\u6e90\u8282\u70b90
    \n for (int i = 1; i <= n; ++i) {
    \n newEdges.emplace_back(0, i, 0);
    \n }<\/p>\n

    schedule.assign(n + 1, INT_MAX);
    \n schedule[0] = 0;<\/p>\n

    \/\/ \u677e\u5f1b\u64cd\u4f5c
    \n for (int i = 0; i < n; ++i) {
    \n for (const Edge& e : newEdges) {
    \n if (schedule[e.u] != INT_MAX && schedule[e.v] > schedule[e.u] + e.weight) {
    \n schedule[e.v] = schedule[e.u] + e.weight;
    \n }
    \n }
    \n }<\/p>\n

    \/\/ \u68c0\u6d4b\u8d1f\u6743\u56de\u8def
    \n for (const Edge& e : newEdges) {
    \n if (schedule[e.u] != INT_MAX && schedule[e.v] > schedule[e.u] + e.weight) {
    \n return false;
    \n }
    \n }<\/p>\n

    return true;
    \n}<\/p>\n

    int main() {
    \n \/\/ \u8bfe\u7a0b\u5b89\u6392\u95ee\u9898
    \n \/\/ \u8bfe\u7a0b: 1(\u7b97\u6cd5), 2(\u6570\u636e\u7ed3\u6784), 3(\u7a0b\u5e8f\u8bbe\u8ba1), 4(\u8ba1\u7b97\u673a\u7f51\u7edc), 5(\u64cd\u4f5c\u7cfb\u7edf)
    \n \/\/ \u7ea6\u675f\u6761\u4ef6:
    \n \/\/ 1. \u6570\u636e\u7ed3\u6784\u5fc5\u987b\u5728\u7b97\u6cd5\u4e4b\u540e\u81f3\u5c111\u5468: x1 + 1 <= x2 \u2192 x2 – x1 >= 1 \u2192 x1 – x2 <= -1
    \n \/\/ 2. \u7b97\u6cd5\u5fc5\u987b\u5728\u7a0b\u5e8f\u8bbe\u8ba1\u4e4b\u540e: x3 <= x1 \u2192 x1 – x3 >= 0 \u2192 x3 – x1 <= 0
    \n \/\/ 3. \u8ba1\u7b97\u673a\u7f51\u7edc\u5fc5\u987b\u5728\u6570\u636e\u7ed3\u6784\u4e4b\u540e\u81f3\u5c112\u5468: x2 + 2 <= x4 \u2192 x4 – x2 >= 2 \u2192 x2 – x4 <= -2
    \n \/\/ 4. \u64cd\u4f5c\u7cfb\u7edf\u5fc5\u987b\u5728\u8ba1\u7b97\u673a\u7f51\u7edc\u4e4b\u540e: x4 <= x5 \u2192 x5 – x4 >= 0 \u2192 x4 – x5 <= 0
    \n \/\/ 5. \u64cd\u4f5c\u7cfb\u7edf\u5fc5\u987b\u5728\u6570\u636e\u7ed3\u6784\u4e4b\u540e\u81f3\u5c113\u5468: x2 + 3 <= x5 \u2192 x5 – x2 >= 3 \u2192 x2 – x5 <= -3
    \n \/\/ 6. \u6240\u6709\u8bfe\u7a0b\u5fc5\u987b\u572810\u5468\u5185\u5b8c\u6210: xi <= 10 \u2192 xi – x0 <= 10 (x0=0)<\/p>\n

    int n = 5; \/\/ 5\u95e8\u8bfe\u7a0b
    \n vector<Edge> edges;
    \n vector<string> courseNames = {"", "\u7b97\u6cd5", "\u6570\u636e\u7ed3\u6784", "\u7a0b\u5e8f\u8bbe\u8ba1", "\u8ba1\u7b97\u673a\u7f51\u7edc", "\u64cd\u4f5c\u7cfb\u7edf"};<\/p>\n

    \/\/ \u6dfb\u52a0\u7ea6\u675f\u5bf9\u5e94\u7684\u8fb9
    \n edges.emplace_back(2, 1, -1); \/\/ \u7ea6\u675f1
    \n edges.emplace_back(1, 3, 0); \/\/ \u7ea6\u675f2
    \n edges.emplace_back(4, 2, -2); \/\/ \u7ea6\u675f3
    \n edges.emplace_back(5, 4, 0); \/\/ \u7ea6\u675f4
    \n edges.emplace_back(5, 2, -3); \/\/ \u7ea6\u675f5<\/p>\n

    \/\/ \u6240\u6709\u8bfe\u7a0b\u5fc5\u987b\u572810\u5468\u5185\u5b8c\u6210
    \n for (int i = 1; i <= n; ++i) {
    \n edges.emplace_back(i, 0, 10); \/\/ xi – x0 <= 10
    \n }<\/p>\n

    vector<int> schedule;
    \n bool possible = solveCourseScheduling(edges, n, schedule);<\/p>\n

    if (possible) {
    \n cout << "\u8bfe\u7a0b\u5b89\u6392\u65b9\u6848:" << endl;
    \n for (int i = 1; i <= n; ++i) {
    \n cout << courseNames[i] << " \u5b89\u6392\u5728\u7b2c " << schedule[i] << " \u5468" << endl;
    \n }
    \n cout << "\u6240\u6709\u8bfe\u7a0b\u5c06\u5728\u7b2c " << schedule[5] << " \u5468\u524d\u5b8c\u6210" << endl;
    \n } else {
    \n cout << "\u65e0\u6cd5\u5b89\u6392\u8fd9\u4e9b\u8bfe\u7a0b，\u5b58\u5728\u51b2\u7a81\u7684\u5148\u4fee\u5173\u7cfb" << endl;
    \n }<\/p>\n

    return 0;
    \n}<\/p>\n

    \"\"<\/h3>\n

    <\/h4>\n

    \u601d\u8003\u9898<\/span><\/h3>\n
  • \n

    \u6bd4\u8f83\u4e0d\u540c\u7b97\u6cd5：\u5bf9\u4e8e\u4e00\u4e2a\u6709 1000 \u4e2a\u8282\u70b9\u548c 10000 \u6761\u8fb9\u7684\u56fe，\u5206\u522b\u8ba8\u8bba\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u9009\u62e9 Bellman-Ford\u3001DAG \u6700\u77ed\u8def\u5f84\u7b97\u6cd5\u6216 Dijkstra \u7b97\u6cd5\u66f4\u5408\u9002\u3002<\/p>\n<\/li>\n

  • \n

    \u8d1f\u6743\u8fb9\u5904\u7406：\u5982\u679c\u56fe\u4e2d\u5b58\u5728\u8d1f\u6743\u8fb9\u4f46\u4e0d\u5b58\u5728\u8d1f\u6743\u56de\u8def，Dijkstra \u7b97\u6cd5\u662f\u5426\u4ecd\u7136\u9002\u7528？\u4e3a\u4ec0\u4e48？\u8bbe\u8ba1\u4e00\u4e2a\u53cd\u4f8b\u6765\u8bf4\u660e\u4f60\u7684\u7ed3\u8bba\u3002<\/p>\n<\/li>\n

  • \n

    \u7b97\u6cd5\u4f18\u5316：\u5982\u4f55\u4f18\u5316 Dijkstra \u7b97\u6cd5\u4f7f\u5176\u5728\u7a00\u758f\u56fe\u4e0a\u7684\u6027\u80fd\u66f4\u597d？\u9664\u4e86\u4f18\u5148\u961f\u5217，\u8fd8\u6709\u5176\u4ed6\u53ef\u80fd\u7684\u4f18\u5316\u65b9\u6cd5\u5417？<\/p>\n<\/li>\n

  • \n

    \u5dee\u5206\u7ea6\u675f\u6269\u5c55：\u5982\u4f55\u5c06\u4e0d\u7b49\u5f0f\u00a0(x_j – x_i \\\\geq w)\u00a0\u8f6c\u5316\u4e3a\u9002\u5408\u5dee\u5206\u7ea6\u675f\u7cfb\u7edf\u7684\u5f62\u5f0f？\u8fd9\u5bf9\u5e94\u56fe\u4e2d\u7684\u4ec0\u4e48\u8fb9？<\/p>\n<\/li>\n

  • \n

    \u6700\u77ed\u8def\u5f84\u6811：\u8bc1\u660e\u6700\u77ed\u8def\u5f84\u6811\u662f\u552f\u4e00\u7684\u5f53\u4e14\u4ec5\u5f53\u5bf9\u4e8e\u6bcf\u4e2a\u8282\u70b9 v，\u4ece\u6e90\u8282\u70b9 s \u5230 v \u7684\u6700\u77ed\u8def\u5f84\u662f\u552f\u4e00\u7684\u3002<\/p>\n<\/li>\n

    \"\"<\/h3>\n

    \u672c\u7ae0\u6ce8\u8bb0<\/span><\/h3>\n

    \u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u57fa\u7840\u95ee\u9898，\u672c\u7ae0\u4ecb\u7ecd\u7684\u7b97\u6cd5\u5404\u6709\u5176\u9002\u7528\u573a\u666f：<\/p>\n

    \"\"<\/p>\n

      \n
    • \n

      Bellman-Ford \u7b97\u6cd5：\u9002\u7528\u4e8e\u5b58\u5728\u8d1f\u6743\u8fb9\u7684\u56fe，\u80fd\u591f\u68c0\u6d4b\u8d1f\u6743\u56de\u8def，\u4f46\u65f6\u95f4\u590d\u6742\u5ea6\u8f83\u9ad8（O (VE)）\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d，\u5176\u961f\u5217\u4f18\u5316\u7248\u672c（SPFA \u7b97\u6cd5）\u66f4\u4e3a\u5e38\u7528\u3002<\/p>\n<\/li>\n

    • \n

      DAG \u6700\u77ed\u8def\u5f84\u7b97\u6cd5：\u662f\u5904\u7406\u6709\u5411\u65e0\u73af\u56fe\u7684\u9ad8\u6548\u7b97\u6cd5（O (V+E)），\u53ef\u4ee5\u5904\u7406\u8d1f\u6743\u8fb9，\u7279\u522b\u9002\u5408\u4efb\u52a1\u8c03\u5ea6\u3001\u4f9d\u8d56\u5206\u6790\u7b49\u573a\u666f\u3002<\/p>\n<\/li>\n

    • \n

      Dijkstra \u7b97\u6cd5：\u662f\u5904\u7406\u975e\u8d1f\u6743\u56fe\u7684\u9ad8\u6548\u7b97\u6cd5，\u4f7f\u7528\u4f18\u5148\u961f\u5217\u4f18\u5316\u540e\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O (ElogV)，\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5bfc\u822a\u7cfb\u7edf\u3001\u7f51\u7edc\u8def\u7531\u7b49\u9886\u57df\u3002<\/p>\n<\/li>\n

    • \n

      \u5dee\u5206\u7ea6\u675f\u7cfb\u7edf：\u5c55\u793a\u4e86\u56fe\u8bba\u4e0e\u7ebf\u6027\u89c4\u5212\u7684\u6df1\u523b\u8054\u7cfb，\u63d0\u4f9b\u4e86\u4e00\u79cd\u6c42\u89e3\u7279\u5b9a\u7c7b\u578b\u7ea6\u675f\u95ee\u9898\u7684\u6709\u6548\u65b9\u6cd5\u3002<\/p>\n<\/li>\n<\/ul>\n

      \"\"<\/p>\n

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      \"\"<\/p>\n

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