如何用c++实现求最大网络流

求最大网络流的经典算法是Ford-Fulkerson算法，它的核心思想是不断寻找增广路径，并更新残留网络，直到不存在增广路径为止。以下是使用C++实现Ford-Fulkerson算法求解最大网络流的示例代码：

#include <iostream>
#include <queue>
#include <cstring>
using namespace std;

#define MAX_VERTEX_NUM 100 // 最大顶点数
#define INF 0x7fffffff // 无穷大

// 定义邻接矩阵结构体
typedef struct {
    int matrix[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; // 邻接矩阵数组
    int vertex_num; // 顶点数
    int edge_num; // 边数
} AdjMatrix;

// 初始化邻接矩阵
void initAdjMatrix(AdjMatrix* graph, int vertex_num) {
    graph->vertex_num = vertex_num;
    graph->edge_num = 0;
    for (int i = 0; i < vertex_num; i++) {
        for (int j = 0; j < vertex_num; j++) {
            graph->matrix[i][j] = 0;
        }
    }
}

// 添加有向边
void addDirectedEdge(AdjMatrix* graph, int from, int to, int weight) {
    graph->matrix[from][to] = weight;
    graph->edge_num++;
}

// Ford-Fulkerson算法求最大网络流
int maxFlow(AdjMatrix* graph, int source, int sink) {
    int n = graph->vertex_num;
    // 创建残留网络
    int residual[MAX_VERTEX_NUM][MAX_VERTEX_NUM];
    memcpy(residual, graph->matrix, sizeof(residual));
    int parent[MAX_VERTEX_NUM];
    int max_flow = 0;
    while (true) {
        // 使用BFS寻找增广路径
        memset(parent, -1, sizeof(parent));
        queue<int> q;
        q.push(source);
        parent[source] = source;
        while (!q.empty()) {
            int u = q.front();
            q.pop();
            for (int v = 0; v < n; v++) {
                if (parent[v] == -1 && residual[u][v] > 0) {
                    parent[v] = u;
                    q.push(v);
                }
            }
        }
        // 如果不存在增广路径，则退出循环
        if (parent[sink] == -1) {
            break;
        }
        // 计算增广路径上的最小流量
        int path_flow = INF;
        for (int v = sink; v != source; v = parent[v]) {
            int u = parent[v];
            path_flow = min(path_flow, residual[u][v]);
        }
        // 更新残留网络，并累加最大流量
        for (int v = sink; v != source; v = parent[v]) {
            int u = parent[v];
            residual[u][v] -= path_flow;
            residual[v][u] += path_flow;
        }
        max_flow += path_flow;
    }
    return max_flow;
}

int main() {
    AdjMatrix graph;
    initAdjMatrix(&graph, 6);
    addDirectedEdge(&graph, 0, 1, 16);
    addDirectedEdge(&graph, 0, 2, 13);
    addDirectedEdge(&graph, 1, 2, 10);
    addDirectedEdge(&graph, 1, 3, 12);
    addDirectedEdge(&graph, 2, 1, 4);
    addDirectedEdge(&graph, 2, 4, 14);
    addDirectedEdge(&graph, 3, 2, 9);
    addDirectedEdge(&graph, 3, 5, 20);
    addDirectedEdge(&graph, 4, 3, 7);
    addDirectedEdge(&graph, 4, 5, 4);
    int source = 0;
    int sink = 5;
    int max_flow = maxFlow(&graph, source, sink);
    cout << "最大网络流为：" << max_flow << endl;
    return 0;
}

以上代码中，使用邻接矩阵存储图的结构，使用Ford-Fulkerson算法求解最大网络流。主函数中构造了一个有向图，然后调用maxFlow函数求解最大网络流，并输出结果。