We strongly recommend to refer below post as a prerequisite.
Hopcroft–Karp Algorithm for Maximum Matching | Set 1 (Introduction)
There are few important things to note before we start implementation.
- We need to find an augmenting path (A path that alternates between matching and not matching edges and has free vertices as starting and ending points).
- Once we find the augmenting path, we need to add the found path to the existing Matching. Here adding a path means, making previous matching edges on this path as not-matching and previous not-matching edges as matching.
The idea is to use BFS (Breadth First Search) to find augmenting paths. Since BFS traverses level by level, it is used to divide the graph in layers of matching and not matching edges. A dummy vertex NIL is added that is connected to all vertices on the left side and all vertices on the right side. The following arrays are used to find augmenting paths. Distance to NIL is initialized as INF (infinite). If we start from a dummy vertex and come back to it using alternating paths of distinct vertices, then there is an augmenting path.
- pairU[]: An array of size m+1 where m is the number of vertices on the left side of the Bipartite Graph. pairU[u] stores paa r of u on the right side if u is matched and NIL otherwise.
- pairV[]: An array of size n+1 where n is several vertices on the right side of the Bipartite Graph. pairV[v] stores a pair of v on the left side if v is matched and NIL otherwise.
- dist[]: An array of size m+1 where m is several vertices on the left side of the Bipartite Graph. dist[u] is initialized as 0 if u is not matching and INF (infinite) otherwise. dist[] of NIL is also initialized as INF
Once an augmenting path is found, DFS (Depth First Search) is used to add augmenting paths to current matching. DFS simply follows the distance array setup by BFS. It fills values in pairU[u] and pairV[v] if v is next to u in BFS.
Below is the implementation of above Hopkroft Karp algorithm.
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;
static class BipGraph
{ int m, n;
List<Integer>[] adj;
int [] pairU, pairV, dist;
int hopcroftKarp()
{
pairU = new int [m + 1 ];
pairV = new int [n + 1 ];
dist = new int [m + 1 ];
Arrays.fill(pairU, NIL);
Arrays.fill(pairV, NIL);
int result = 0 ;
while (bfs())
{
for ( int u = 1 ; u <= m; u++)
if (pairU[u] == NIL && dfs(u))
result++;
}
return result;
}
boolean bfs()
{
Queue<Integer> Q = new LinkedList<>();
for ( int u = 1 ; u <= m; u++)
{
if (pairU[u] == NIL)
{
dist[u] = 0 ;
Q.add(u);
}
else
dist[u] = INF;
}
dist[NIL] = INF;
while (!Q.isEmpty())
{
int u = Q.poll();
if (dist[u] < dist[NIL])
{
for ( int i : adj[u])
{
int v = i;
if (dist[pairV[v]] == INF)
{
dist[pairV[v]] = dist[u] + 1 ;
Q.add(pairV[v]);
}
}
}
}
return (dist[NIL] != INF);
}
boolean dfs( int u)
{
if (u != NIL)
{
for ( int i : adj[u])
{
int v = i;
if (dist[pairV[v]] == dist[u] + 1 )
{
if (dfs(pairV[v]) == true )
{
pairV[v] = u;
pairU[u] = v;
return true ;
}
}
}
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<>());
}
void addEdge( int u, int v)
{
adj[u].add(v);
}
} 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());
} } |
// GFG // C# code for this approach using System;
using System.Collections.Generic;
using System.Linq;
class HopcroftKarp
{ const int NIL = 0;
const int INF = int .MaxValue;
static void Main()
{
int n = 4; // number of nodes in set U
int m = 4; // number of nodes in set V
var g = new BipGraph(n, m);
g.addEdge(1, 2);
g.addEdge(1, 3);
g.addEdge(2, 1);
g.addEdge(3, 2);
g.addEdge(4, 2);
g.addEdge(4, 4);
Console.WriteLine( "Size of maximum matching is " +
g.hopcroftKarp());
}
class BipGraph
{
private readonly int m;
private readonly int n;
private readonly List< int >[] adj;
private int [] pairU;
private int [] pairV;
private int [] dist;
public BipGraph( int m, int n)
{
this .m = m;
this .n = n;
adj = new List< int >[m + 1];
for ( int i = 0; i <= m; i++)
{
adj[i] = new List< int >();
}
}
public void addEdge( int u, int v)
{
adj[u].Add(v);
}
public int hopcroftKarp()
{
pairU = Enumerable.Repeat(NIL, m + 1).ToArray();
pairV = Enumerable.Repeat(NIL, n + 1).ToArray();
dist = Enumerable.Repeat(0, m + 1).ToArray();
int result = 0;
while (bfs())
{
for ( int u = 1; u <= m; u++)
{
if (pairU[u] == NIL && dfs(u))
{
result++;
}
}
}
return result;
}
private bool bfs()
{
var Q = new Queue< int >();
for ( int u = 1; u <= m; u++)
{
if (pairU[u] == NIL)
{
dist[u] = 0;
Q.Enqueue(u);
}
else
{
dist[u] = INF;
}
}
dist[NIL] = INF;
while (Q.Count > 0)
{
int u = Q.Dequeue();
if (dist[u] < dist[NIL])
{
foreach ( int v in adj[u])
{
if (dist[pairV[v]] == INF)
{
dist[pairV[v]] = dist[u] + 1;
Q.Enqueue(pairV[v]);
}
}
}
}
return dist[NIL] != INF;
}
private bool dfs( int u)
{
if (u != NIL)
{
foreach ( int v in adj[u])
{
if (dist[pairV[v]] == dist[u] + 1)
{
if (dfs(pairV[v]))
{
pairV[v] = u;
pairU[u] = v;
return true ;
}
}
}
dist[u] = INF;
return false ;
}
return true ;
}
}
} // Thic is written by Sundaram |
// Javascript implementation of Hopcroft Karp algorithm for maximum matching class BipGraph { constructor(m, n) {
this .__m = m;
this .__n = n;
this .__adj = [...Array(m + 1)].map(() => []);
}
addEdge(u, v) {
this .__adj[u].push(v); // Add u to v’s list.
}
bfs() {
const Q = [];
for (let u = 1; u <= this .__m; u++) {
if ( this .__pairU[u] === NIL) {
this .__dist[u] = 0;
Q.push(u);
} else {
this .__dist[u] = INF;
}
}
this .__dist[NIL] = INF;
while (Q.length > 0) {
const u = Q.shift();
if ( this .__dist[u] < this .__dist[NIL]) {
for (const v of this .__adj[u]) {
if ( this .__dist[ this .__pairV[v]] === INF) {
this .__dist[ this .__pairV[v]] = this .__dist[u] + 1;
Q.push( this .__pairV[v]);
}
}
}
}
return this .__dist[NIL] !== INF;
}
dfs(u) {
if (u !== NIL) {
for (const v of this .__adj[u]) {
if ( this .__dist[ this .__pairV[v]] === this .__dist[u] + 1) {
if ( this .dfs( this .__pairV[v])) {
this .__pairV[v] = u;
this .__pairU[u] = v;
return true ;
}
}
}
this .__dist[u] = INF;
return false ;
}
return true ;
}
hopcroftKarp() {
this .__pairU = Array( this .__m + 1).fill(0);
this .__pairV = Array( this .__n + 1).fill(0);
this .__dist = Array( this .__m + 1).fill(0);
let result = 0;
while ( this .bfs()) {
for (let u = 1; u <= this .__m; u++) {
if ( this .__pairU[u] === NIL && this .dfs(u)) {
result++;
}
}
}
return result;
}
} const INF = 2147483647; const NIL = 0; // Driver Program const 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); console.log(`Size of maximum matching is ${g.hopcroftKarp()}`); |
// C++ implementation of Hopcroft Karp algorithm for // maximum matching #include<bits/stdc++.h> 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< int > *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< int > 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< int >::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< int >::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< int >[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;
} |
# Python3 implementation of Hopcroft Karp algorithm for # maximum matching from queue import Queue
INF = 2147483647
NIL = 0
# A class to represent Bipartite graph for Hopcroft # 3 Karp implementation class BipGraph( object ):
# Constructor
def __init__( self , m, n):
# m and n are number of vertices on left
# and right sides of Bipartite Graph
self .__m = m
self .__n = 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
self .__adj = [[] for _ in range (m + 1 )]
# To add edge from u to v and v to u
def addEdge( self , u, v):
self .__adj[u].append(v) # Add u to v’s list.
# Returns true if there is an augmenting path, else returns
# false
def bfs( self ):
Q = Queue()
# First layer of vertices (set distance as 0)
for u in range ( 1 , self .__m + 1 ):
# If this is a free vertex, add it to queue
if self .__pairU[u] = = NIL:
# u is not matched3
self .__dist[u] = 0
Q.put(u)
# Else set distance as infinite so that this vertex
# is considered next time
else :
self .__dist[u] = INF
# Initialize distance to NIL as infinite
self .__dist[NIL] = INF
# Q is going to contain vertices of left side only.
while not Q.empty():
# Dequeue a vertex
u = Q.get()
# If this node is not NIL and can provide a shorter path to NIL
if self .__dist[u] < self .__dist[NIL]:
# Get all adjacent vertices of the dequeued vertex u
for v in self .__adj[u]:
# If pair of v is not considered so far
# (v, pairV[V]) is not yet explored edge.
if self .__dist[ self .__pairV[v]] = = INF:
# Consider the pair and add it to queue
self .__dist[ self .__pairV[v]] = self .__dist[u] + 1
Q.put( self .__pairV[v])
# If we could come back to NIL using alternating path of distinct
# vertices then there is an augmenting path
return self .__dist[NIL] ! = INF
# Returns true if there is an augmenting path beginning with free vertex u
def dfs( self , u):
if u ! = NIL:
# Get all adjacent vertices of the dequeued vertex u
for v in self .__adj[u]:
if self .__dist[ self .__pairV[v]] = = self .__dist[u] + 1 :
# If dfs for pair of v also returns true
if self .dfs( self .__pairV[v]):
self .__pairV[v] = u
self .__pairU[u] = v
return True
# If there is no augmenting path beginning with u.
self .__dist[u] = INF
return False
return True
def hopcroftKarp( self ):
# 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
self .__pairU = [ 0 for _ in range ( self .__m + 1 )]
# pairV[v] stores pair of v in matching. If v
# doesn't have any pair, then pairU[v] is NIL
self .__pairV = [ 0 for _ in range ( self .__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
self .__dist = [ 0 for _ in range ( self .__m + 1 )]
# Initialize result
result = 0
# Keep updating the result while there is an
# augmenting path.
while self .bfs():
# Find a free vertex
for u in range ( 1 , self .__m + 1 ):
# If current vertex is free and there is
# an augmenting path from current vertex
if self .__pairU[u] = = NIL and self .dfs(u):
result + = 1
return result
# Driver Program if __name__ = = "__main__" :
g = 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 )
print ( "Size of maximum matching is %d" % g.hopcroftKarp())
|
Size of maximum matching is 4
Time Complexity : O(√V x E), where E is the number of Edges and V is the number of vertices.
Auxiliary Space : O(V) as we are using extra space for storing u and v.
The above implementation is mainly adopted from the algorithm provided on Wiki page of Hopcroft Karp algorithm.