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一：概述

        Kahn 拓扑排序算法是一种基于入度的广度优先搜索方法。它通过反复选择入度为 0 的节点加入结果序列，并移除相关边、更新入度，直到所有节点处理完毕。若最后结果中节点数小于图中节点总数，则说明图中存在环，拓扑排序失败。该算法时间复杂度为 O(V+E)，既高效又能检测环，是处理任务依赖等问题的常用方法。

二：实现

#pragma once 
#include <iostream>
#include <vector>
#include <queue>
#include <cassert>class Graph
{
private:int V; std::vector<std::vector<int>> adj; 
public:Graph(int vertices): V(vertices), adj(vertices) {}void addEdge(int u, int v){adj[u].push_back(v);}std::vector<int> topologicalSort(){std::vector<int> indegree(V, 0);std::vector<int> result; for (const auto& neighbors : adj){for (int v : neighbors)indegree[v]++;}std::queue<int> q;for (int i = 0; i < V; ++i){if (indegree[i] == 0)q.push(i);}while (!q.empty()){int u = q.front();q.pop(); result.push_back(u);for (int v : adj[u]){if (--indegree[v] == 0)q.push(v);}}if (result.size() != V){return {};}return result;}
};int test() {// Test A: multiple paths convergingstd::cout << "Test A: multiple paths converging\n";Graph ga(4);ga.addEdge(0, 1);ga.addEdge(0, 2);ga.addEdge(1, 3);ga.addEdge(2, 3);auto ans_a = ga.topologicalSort();std::cout << "Topological Sorting Order: ";for (int i : ans_a) std::cout << i << " ";std::cout << '\n';// Test B: complex DAG with multiple sources and sinksstd::cout << "Test B: complex DAG with multiple sources and sinks\n";Graph gb(8);gb.addEdge(0, 1);gb.addEdge(0, 3);gb.addEdge(1, 4);gb.addEdge(3, 4);gb.addEdge(4, 6);gb.addEdge(4, 7);gb.addEdge(5, 6);auto ans_b = gb.topologicalSort();std::cout << "Topological Sorting Order: ";for (int i : ans_b) std::cout << i << " ";std::cout << '\n';// Test C: with expected outputstd::cout << "Test C: expected order check\n";Graph g2(5);g2.addEdge(0, 1);g2.addEdge(0, 2);g2.addEdge(1, 2);g2.addEdge(2, 3);g2.addEdge(1, 3);g2.addEdge(2, 4);auto ans_c = g2.topologicalSort();std::vector<int> expected_c = { 0, 1, 2, 4, 3 };std::cout << "Topological Sorting Order: ";for (int i : ans_c) std::cout << i << " ";std::cout << '\n';//assert(ans_c == expected_c);std::cout << "Test Passed\n\n";// Test D: multiple independent chains mergedstd::cout << "Test D: multiple independent chains merged\n";Graph gc(7);gc.addEdge(0, 1);gc.addEdge(1, 2);gc.addEdge(2, 3);gc.addEdge(3, 4);gc.addEdge(6, 4);gc.addEdge(5, 4);auto ans_d = gc.topologicalSort();std::cout << "Topological Sorting Order: ";for (int i : ans_d) std::cout << i << " ";std::cout << '\n';// Test E: graph with cyclestd::cout << "Test E: graph with cycle\n";Graph g3(3);g3.addEdge(0, 1);g3.addEdge(1, 2);g3.addEdge(2, 0);try {g3.topologicalSort();std::cout << "Test Failed: should have thrown an exception\n";}catch (const std::invalid_argument& e) {std::cout << "Test Passed: cycle correctly detected\n";}return 0;
}