使用push-relabel

Question

我正在尝试运行push relabel算法来获得最小s-t削减。

我从https://www.geeksforgeeks.org/push-relabel-algorithm-set-2-implementation/取了算法

// C++ program to implement push-relabel algorithm for 
// getting maximum flow of graph 
#include <bits/stdc++.h> 
using namespace std; 

struct Edge 
{ 
    // To store current flow and capacity of edge 
    int flow, capacity; 

    // An edge u--->v has start vertex as u and end 
    // vertex as v. 
    int u, v; 

    Edge(int flow, int capacity, int u, int v) 
    { 
        this->flow = flow; 
        this->capacity = capacity; 
        this->u = u; 
        this->v = v; 
    } 
}; 

// Represent a Vertex 
struct Vertex 
{ 
    int h, e_flow; 

    Vertex(int h, int e_flow) 
    { 
        this->h = h; 
        this->e_flow = e_flow; 
    } 
}; 

// To represent a flow network 
class Graph 
{ 
    int V; // No. of vertices 
    vector<Vertex> ver; 
    vector<Edge> edge; 

    // Function to push excess flow from u 
    bool push(int u); 

    // Function to relabel a vertex u 
    void relabel(int u); 

    // This function is called to initialize 
    // preflow 
    void preflow(int s); 

    // Function to reverse edge 
    void updateReverseEdgeFlow(int i, int flow); 

public: 
    Graph(int V); // Constructor 

    // function to add an edge to graph 
    void addEdge(int u, int v, int w); 

    // returns maximum flow from s to t 
    int getMaxFlow(int s, int t); 
}; 

Graph::Graph(int V) 
{ 
    this->V = V; 

    // all vertices are initialized with 0 height 
    // and 0 excess flow 
    for (int i = 0; i < V; i++) 
        ver.push_back(Vertex(0, 0)); 
} 

void Graph::addEdge(int u, int v, int capacity) 
{ 
    // flow is initialized with 0 for all edge 
    edge.push_back(Edge(0, capacity, u, v)); 
} 

void Graph::preflow(int s) 
{ 
    // Making h of source Vertex equal to no. of vertices 
    // Height of other vertices is 0. 
    ver[s].h = ver.size(); 

    // 
    for (int i = 0; i < edge.size(); i++) 
    { 
        // If current edge goes from source 
        if (edge[i].u == s) 
        { 
            // Flow is equal to capacity 
            edge[i].flow = edge[i].capacity; 

            // Initialize excess flow for adjacent v 
            ver[edge[i].v].e_flow += edge[i].flow; 

            // Add an edge from v to s in residual graph with 
            // capacity equal to 0 
            edge.push_back(Edge(-edge[i].flow, 0, edge[i].v, s)); 
        } 
    } 
} 

// returns index of overflowing Vertex 
int overFlowVertex(vector<Vertex>& ver) 
{ 
    for (int i = 1; i < ver.size() - 1; i++) 
    if (ver[i].e_flow > 0) 
            return i; 

    // -1 if no overflowing Vertex 
    return -1; 
} 

// Update reverse flow for flow added on ith Edge 
void Graph::updateReverseEdgeFlow(int i, int flow) 
{ 
    int u = edge[i].v, v = edge[i].u; 

    for (int j = 0; j < edge.size(); j++) 
    { 
        if (edge[j].v == v && edge[j].u == u) 
        { 
            edge[j].flow -= flow; 
            return; 
        } 
    } 

    // adding reverse Edge in residual graph 
    Edge e = Edge(0, flow, u, v); 
    edge.push_back(e); 
} 

// To push flow from overflowing vertex u 
bool Graph::push(int u) 
{ 
    // Traverse through all edges to find an adjacent (of u) 
    // to which flow can be pushed 
    for (int i = 0; i < edge.size(); i++) 
    { 
        // Checks u of current edge is same as given 
        // overflowing vertex 
        if (edge[i].u == u) 
        { 
            // if flow is equal to capacity then no push 
            // is possible 
            if (edge[i].flow == edge[i].capacity) 
                continue; 

            // Push is only possible if height of adjacent 
            // is smaller than height of overflowing vertex 
            if (ver[u].h > ver[edge[i].v].h) 
            { 
                // Flow to be pushed is equal to minimum of 
                // remaining flow on edge and excess flow. 
                int flow = min(edge[i].capacity - edge[i].flow, 
                            ver[u].e_flow); 

                // Reduce excess flow for overflowing vertex 
                ver[u].e_flow -= flow; 

                // Increase excess flow for adjacent 
                ver[edge[i].v].e_flow += flow; 

                // Add residual flow (With capacity 0 and negative 
                // flow) 
                edge[i].flow += flow; 

                updateReverseEdgeFlow(i, flow); 

                return true; 
            } 
        } 
    } 
    return false; 
} 

// function to relabel vertex u 
void Graph::relabel(int u) 
{ 
    // Initialize minimum height of an adjacent 
    int mh = INT_MAX; 

    // Find the adjacent with minimum height 
    for (int i = 0; i < edge.size(); i++) 
    { 
        if (edge[i].u == u) 
        { 
            // if flow is equal to capacity then no 
            // relabeling 
            if (edge[i].flow == edge[i].capacity) 
                continue; 

            // Update minimum height 
            if (ver[edge[i].v].h < mh) 
            { 
                mh = ver[edge[i].v].h; 

                // updating height of u 
                ver[u].h = mh + 1; 
            } 
        } 
    } 
} 

// main function for printing maximum flow of graph 
int Graph::getMaxFlow(int s, int t) 
{ 
    preflow(s); 

    // loop untill none of the Vertex is in overflow 
    while (overFlowVertex(ver) != -1) 
    { 
        int u = overFlowVertex(ver); 
        if (!push(u)) 
            relabel(u); 
    } 

    // ver.back() returns last Vertex, whose 
    // e_flow will be final maximum flow 
    return ver.back().e_flow; 
} 

// Driver program to test above functions 
int main() 
{ 
    int V = 6; 
    Graph g(V); 

    // Creating above shown flow network 
    g.addEdge(0, 1, 16); 
    g.addEdge(0, 2, 13); 
    g.addEdge(1, 2, 10); 
    g.addEdge(2, 1, 4); 
    g.addEdge(1, 3, 12); 
    g.addEdge(2, 4, 14); 
    g.addEdge(3, 2, 9); 
    g.addEdge(3, 5, 20); 
    g.addEdge(4, 3, 7); 
    g.addEdge(4, 5, 4); 

    // Initialize source and sink 
    int s = 0, t = 5; 

    cout << "Maximum flow is " << g.getMaxFlow(s, t); 
    return 0; 
} 

该算法工作正常，但仅给出了最终的最大流量。我四处搜寻以获取最小切割，发现我们需要在残差图中找到可以从源访问的所有顶点，然后找到最小切割。虽然有道理，但上述算法并未给出最终的残差图，我想我应该从所有边缘创建最终的残差图，这对于大型图而言似乎在计算上非常昂贵。有没有办法在使用push-relabel同时计算最大流量的同时计算最小切割？