推重标签算法 |第 2 组（实施）
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推重标签算法 |第 1 套（介绍和插图）
问题陈述：给定一个表示流网络的图，其中每条边都有容量。同样给定图中的两个顶点source 's' 和sink 't'，找到从 s 到 t 的最大可能流，具有以下约束：
a)边上的流量不超过边的给定容量。
b)对于除 s 和 t 之外的每个顶点，流入流等于流出流。
例如，考虑 CLRS 书中的下图。
_0.png)
上图中的最大可能流量为 23。
_1.png)
Push-Relabel Algorithm
1) Initialize PreFlow : Initialize Flows and Heights
2) While it is possible to perform a Push() or Relabel() 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.以下是 Push Relabel 算法中执行的主要操作。
Push-Relabel算法主要有3个操作
1. Initialize 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()用于从有多余流量的节点产生流量。如果一个顶点有多余的流量并且有一个高度较小的相邻节点（在残差图中），我们将流量从顶点推到高度较低的相邻节点。通过管道（边缘）的推动流量等于边缘的过量流量和容量的最小值。
3. Relabel()操作用于当一个顶点有多余的流量并且它的相邻顶点都不在较低的高度时。我们基本上增加了顶点的高度，以便我们可以执行 push()。为了增加高度，我们选择最小高度相邻（在残差图中，即我们可以添加流量的相邻）并将其加 1。
执行
以下实现使用以下结构来表示流网络。
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
}下面的代码使用给定的图本身作为流网络和残差图。我们没有为残差图创建单独的图，并且为了简单起见使用了相同的图。
C++
// C++ program to implement push-relabel algorithm for
// getting maximum flow of graph
#include
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 ver;
vector 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& 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 until 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;
} 输出：
Maximum flow is 23