最大匹配的 Hopcroft–Karp 算法第 2 组（实施）
我们强烈建议您参考以下帖子作为先决条件。
最大匹配的 Hopcroft–Karp 算法第 1 套（介绍）
在我们开始实施之前，有几件重要的事情需要注意。
- 我们需要找到一条增广路径（一条在匹配和不匹配边之间交替的路径，并且有自由顶点作为起点和终点）。
- 一旦我们找到交替路径，我们需要将找到的路径添加到现有的 Matching中。这里添加路径的意思是，使该路径上之前匹配的边为不匹配，之前不匹配的边为匹配。
这个想法是使用 BFS（广度优先搜索）来寻找增广路径。由于 BFS 是逐层遍历的，因此它用于将图划分为匹配边和不匹配边的层。添加了一个虚拟顶点 NIL，它连接到左侧的所有顶点和右侧的所有顶点。以下数组用于查找增广路径。到 NIL 的距离被初始化为 INF（无限）。如果我们从虚拟顶点开始并使用不同顶点的交替路径返回它，那么就有一条增广路径。
- pairU[]：大小为 m+1 的数组，其中 m 是二分图左侧的顶点数。如果 u 匹配，pairU[u] 将 u 对存储在右侧，否则存储 NIL。
- pairV[]：大小为 n+1 的数组，其中 n 是二分图右侧的顶点数。如果 v 匹配，pairV[v] 将 v 对存储在左侧，否则存储 NIL。
- dist[]：大小为 m+1 的数组，其中 m 是二分图左侧的顶点数。如果 u 不匹配，则 dist[u] 初始化为 0，否则初始化为 INF（无限）。 NIL 的 dist[] 也被初始化为 INF
一旦找到增广路径，就使用 DFS（深度优先搜索）将增广路径添加到当前匹配中。 DFS 只是遵循 BFS 设置的距离数组。如果在 BFS 中 v 与 u 相邻，它会填充 pairU[u] 和 pairV[v] 中的值。
下面是上述 Hopkroft Karp 算法的实现。
C++14
// C++ implementation of Hopcroft Karp algorithm for
// maximum matching
#include
using namespace std;
#define NIL 0
#define INF INT_MAX
// A class to represent Bipartite graph for Hopcroft
// Karp implementation
class BipGraph
{
// m and n are number of vertices on left
// and right sides of Bipartite Graph
int m, n;
// adj[u] stores adjacents of left side
// vertex 'u'. The value of u ranges from 1 to m.
// 0 is used for dummy vertex
list *adj;
// These are basically pointers to arrays needed
// for hopcroftKarp()
int *pairU, *pairV, *dist;
public:
BipGraph(int m, int n); // Constructor
void addEdge(int u, int v); // To add edge
// Returns true if there is an augmenting path
bool bfs();
// Adds augmenting path if there is one beginning
// with u
bool dfs(int u);
// Returns size of maximum matching
int hopcroftKarp();
};
// Returns size of maximum matching
int BipGraph::hopcroftKarp()
{
// pairU[u] stores pair of u in matching where u
// is a vertex on left side of Bipartite Graph.
// If u doesn't have any pair, then pairU[u] is NIL
pairU = new int[m+1];
// pairV[v] stores pair of v in matching. If v
// doesn't have any pair, then pairU[v] is NIL
pairV = new int[n+1];
// dist[u] stores distance of left side vertices
// dist[u] is one more than dist[u'] if u is next
// to u'in augmenting path
dist = new int[m+1];
// Initialize NIL as pair of all vertices
for (int u=0; u<=m; u++)
pairU[u] = NIL;
for (int v=0; v<=n; v++)
pairV[v] = NIL;
// Initialize result
int result = 0;
// Keep updating the result while there is an
// augmenting path.
while (bfs())
{
// Find a free vertex
for (int u=1; u<=m; u++)
// If current vertex is free and there is
// an augmenting path from current vertex
if (pairU[u]==NIL && dfs(u))
result++;
}
return result;
}
// Returns true if there is an augmenting path, else returns
// false
bool BipGraph::bfs()
{
queue Q; //an integer queue
// First layer of vertices (set distance as 0)
for (int u=1; u<=m; u++)
{
// If this is a free vertex, add it to queue
if (pairU[u]==NIL)
{
// u is not matched
dist[u] = 0;
Q.push(u);
}
// Else set distance as infinite so that this vertex
// is considered next time
else dist[u] = INF;
}
// Initialize distance to NIL as infinite
dist[NIL] = INF;
// Q is going to contain vertices of left side only.
while (!Q.empty())
{
// Dequeue a vertex
int u = Q.front();
Q.pop();
// If this node is not NIL and can provide a shorter path to NIL
if (dist[u] < dist[NIL])
{
// Get all adjacent vertices of the dequeued vertex u
list::iterator i;
for (i=adj[u].begin(); i!=adj[u].end(); ++i)
{
int v = *i;
// If pair of v is not considered so far
// (v, pairV[V]) is not yet explored edge.
if (dist[pairV[v]] == INF)
{
// Consider the pair and add it to queue
dist[pairV[v]] = dist[u] + 1;
Q.push(pairV[v]);
}
}
}
}
// If we could come back to NIL using alternating path of distinct
// vertices then there is an augmenting path
return (dist[NIL] != INF);
}
// Returns true if there is an augmenting path beginning with free vertex u
bool BipGraph::dfs(int u)
{
if (u != NIL)
{
list::iterator i;
for (i=adj[u].begin(); i!=adj[u].end(); ++i)
{
// Adjacent to u
int v = *i;
// Follow the distances set by BFS
if (dist[pairV[v]] == dist[u]+1)
{
// If dfs for pair of v also returns
// true
if (dfs(pairV[v]) == true)
{
pairV[v] = u;
pairU[u] = v;
return true;
}
}
}
// If there is no augmenting path beginning with u.
dist[u] = INF;
return false;
}
return true;
}
// Constructor
BipGraph::BipGraph(int m, int n)
{
this->m = m;
this->n = n;
adj = new list[m+1];
}
// To add edge from u to v and v to u
void BipGraph::addEdge(int u, int v)
{
adj[u].push_back(v); // Add u to v’s list.
}
// Driver Program
int main()
{
BipGraph g(4, 4);
g.addEdge(1, 2);
g.addEdge(1, 3);
g.addEdge(2, 1);
g.addEdge(3, 2);
g.addEdge(4, 2);
g.addEdge(4, 4);
cout << "Size of maximum matching is " << g.hopcroftKarp();
return 0;
} Java
// Java implementation of Hopcroft Karp
// algorithm for maximum matching
import java.util.ArrayList;
import java.util.Arrays;
import java.util.LinkedList;
import java.util.List;
import java.util.Queue;
class GFG{
static final int NIL = 0;
static final int INF = Integer.MAX_VALUE;
// A class to represent Bipartite graph
// for Hopcroft Karp implementation
static class BipGraph
{
// m and n are number of vertices on left
// and right sides of Bipartite Graph
int m, n;
// adj[u] stores adjacents of left side
// vertex 'u'. The value of u ranges
// from 1 to m. 0 is used for dummy vertex
List[] adj;
// These are basically pointers to arrays
// needed for hopcroftKarp()
int[] pairU, pairV, dist;
// Returns size of maximum matching
int hopcroftKarp()
{
// pairU[u] stores pair of u in matching where u
// is a vertex on left side of Bipartite Graph.
// If u doesn't have any pair, then pairU[u] is NIL
pairU = new int[m + 1];
// pairV[v] stores pair of v in matching. If v
// doesn't have any pair, then pairU[v] is NIL
pairV = new int[n + 1];
// dist[u] stores distance of left side vertices
// dist[u] is one more than dist[u'] if u is next
// to u'in augmenting path
dist = new int[m + 1];
// Initialize NIL as pair of all vertices
Arrays.fill(pairU, NIL);
Arrays.fill(pairV, NIL);
// Initialize result
int result = 0;
// Keep updating the result while
// there is an augmenting path.
while (bfs())
{
// Find a free vertex
for(int u = 1; u <= m; u++)
// If current vertex is free and there is
// an augmenting path from current vertex
if (pairU[u] == NIL && dfs(u))
result++;
}
return result;
}
// Returns true if there is an augmenting
// path, else returns false
boolean bfs()
{
// An integer queue
Queue Q = new LinkedList<>();
// First layer of vertices (set distance as 0)
for(int u = 1; u <= m; u++)
{
// If this is a free vertex,
// add it to queue
if (pairU[u] == NIL)
{
// u is not matched
dist[u] = 0;
Q.add(u);
}
// Else set distance as infinite
// so that this vertex is
// considered next time
else
dist[u] = INF;
}
// Initialize distance to
// NIL as infinite
dist[NIL] = INF;
// Q is going to contain vertices
// of left side only.
while (!Q.isEmpty())
{
// Dequeue a vertex
int u = Q.poll();
// If this node is not NIL and
// can provide a shorter path to NIL
if (dist[u] < dist[NIL])
{
// Get all adjacent vertices of
// the dequeued vertex u
for(int i : adj[u])
{
int v = i;
// If pair of v is not considered
// so far (v, pairV[V]) is not yet
// explored edge.
if (dist[pairV[v]] == INF)
{
// Consider the pair and add
// it to queue
dist[pairV[v]] = dist[u] + 1;
Q.add(pairV[v]);
}
}
}
}
// If we could come back to NIL using
// alternating path of distinct vertices
// then there is an augmenting path
return (dist[NIL] != INF);
}
// Returns true if there is an augmenting
// path beginning with free vertex u
boolean dfs(int u)
{
if (u != NIL)
{
for(int i : adj[u])
{
// Adjacent to u
int v = i;
// Follow the distances set by BFS
if (dist[pairV[v]] == dist[u] + 1)
{
// If dfs for pair of v also returns
// true
if (dfs(pairV[v]) == true)
{
pairV[v] = u;
pairU[u] = v;
return true;
}
}
}
// If there is no augmenting path
// beginning with u.
dist[u] = INF;
return false;
}
return true;
}
// Constructor
@SuppressWarnings("unchecked")
public BipGraph(int m, int n)
{
this.m = m;
this.n = n;
adj = new ArrayList[m + 1];
Arrays.fill(adj, new ArrayList<>());
}
// To add edge from u to v and v to u
void addEdge(int u, int v)
{
// Add u to v’s list.
adj[u].add(v);
}
}
// Driver code
public static void main(String[] args)
{
BipGraph g = new BipGraph(4, 4);
g.addEdge(1, 2);
g.addEdge(1, 3);
g.addEdge(2, 1);
g.addEdge(3, 2);
g.addEdge(4, 2);
g.addEdge(4, 4);
System.out.println("Size of maximum matching is " +
g.hopcroftKarp());
}
}
// This code is contributed by sanjeev2552 输出：
Size of maximum matching is 4