# 计算最大流：Push Relabel算法|S2（算法代码实现）

2021年3月15日09:51:46 发表评论 1,100 次浏览

Push-Relabel算法|S1(简介和插图)

a)边缘的流量不超过边缘的给定容量。

b)除s和t以外, 每个顶点的流入流量等于流出流量。

``````Push-Relabel Algorithm
1) Initialize PreFlow : Initialize Flows and Heights

2) While it is possible to perform a Push() or Relablel() on a vertex
// Or while there is a vertex that has excess flow
Do Push() or Relabel()

// At this point all vertices have Excess Flow as 0 (Except source
// and sink)
3) Return flow.``````

Push-Relabel算法中有三个主要操作

1. 初始化PreFlow()它初始化所有顶点的高度和流。
```Preflow()
1) Initialize height and flow of every vertex as 0.
2) Initialize height of source vertex equal to total
number of vertices in graph.
3) Initialize flow of every edge as 0.
4) For all vertices adjacent to source s, flow and
excess flow is equal to capacity initially.```
2. push()用于从流量过大的节点产生流量。如果顶点有多余的流动, 并且有一个相邻的高度较小(在残差图中), 我们将流动从顶点推到相邻的较低高度。通过管道(边缘)的推动流量等于多余流量和边缘容量的最小值。
3. Relabel()当顶点有多余的流量且相邻的顶点都不处于较低高度时, 将使用操作。我们基本上增加了顶点的高度, 以便可以执行push()。为了增加高度, 我们选择相邻的最小高度(在残差图中, 即可以添加流量的相邻高度)并对其加1。

``````struct Vertex
{
int h;      // Height of node
int e_flow; // Excess Flow
}``````
``````struct Edge
{
int u, v; // Edge is from u to v
int flow; // Current flow
int capacity;
}``````
``````class Graph
{
Edge edge[];   // Array of edges
Vertex ver[];  // Array of vertices
}``````

``````// 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展开.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

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

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

``Maximum flow is 23``