hopcroft_karp.cpp File Reference

Implementation of Hopcroft–Karp algorithm. More...

#include <iostream>
#include <cstdlib>
#include <queue>
#include <list>
#include <climits>
#include <memory>
#include <cassert>

Include dependency graph for hopcroft_karp.cpp:

Classes
class	graph::HKGraph
	Represents Bipartite graph for Hopcroft Karp implementation. More...

Namespaces
	graph
	Graph Algorithms.

Functions
void	tests ()
int	main ()
	Main function. More...

Detailed Description

Implementation of Hopcroft–Karp algorithm.

The Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph and produces as output a maximum cardinality matching, it runs in O(E√V) time in worst case.

Bipartite graph

A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

Matching and Not-Matching edges

Given a matching M, edges that are part of matching are called Matching edges and edges that are not part of M (or connect free nodes) are called Not-Matching edges.

Maximum cardinality matching

Given a bipartite graphs G = ( V = ( X , Y ) , E ) whose partition has the parts X and Y, with E denoting the edges of the graph, the goal is to find a matching with as many edges as possible. Equivalently, a matching that covers as many vertices as possible.

Augmenting paths

Given a matching M, an augmenting path is an alternating path that starts from and ends on free vertices. All single edge paths that start and end with free vertices are augmenting paths.

Concept

A matching M is not maximum if there exists an augmenting path. It is also true other way, i.e, a matching is maximum if no augmenting path exists.

Algorithm

1) Initialize the Maximal Matching M as empty. 2) While there exists an Augmenting Path P Remove matching edges of P from M and add not-matching edges of P to M (This increases size of M by 1 as P starts and ends with a free vertex i.e. a node that is not part of matching.) 3) Return M.

Author

Krishna Pal Deora

Function Documentation

main()

int main

(

void

)

Main function.

Returns

0 on exit

{
    tests();  // perform self-tests
 
    int v1 = 0, v2 = 0, e = 0;
    std::cin >> v1 >> v2 >> e; // vertices of left side, right side and edges
    HKGraph g(v1, v2);  
    int u = 0, v = 0;
    for (int i = 0; i < e; ++i)
    {
        std::cin >> u >> v;
        g.addEdge(u, v);
    }
  
    int res = g.hopcroftKarpAlgorithm();
    std::cout << "Maximum matching is " << res <<"\n";
 
    return 0;
 
}

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tests()

void tests

(

)

Self-test implementation

Returns

none

            {
     // Sample test case 1
         int v1a = 3, v1b = 5, e1 = 2;  // vertices of left side, right side and edges
         HKGraph g1(v1a, v1b); // execute the algorithm 
 
         g1.addEdge(0,1);
         g1.addEdge(1,4);
 
         int expected_res1 = 0; // for the above sample data, this is the expected output
         int res1 = g1.hopcroftKarpAlgorithm();
 
         assert(res1 == expected_res1); // assert check to ensure that the algorithm executed correctly for test 1
    
     // Sample test case 2
             int v2a = 4, v2b = 4, e2 = 6;  // vertices of left side, right side and edges
         HKGraph g2(v2a, v2b); // execute the algorithm 
 
             g2.addEdge(1,1);
         g2.addEdge(1,3);
         g2.addEdge(2,3);
         g2.addEdge(3,4);
         g2.addEdge(4,3);
             g2.addEdge(4,2);
    
         int expected_res2 = 0; // for the above sample data, this is the expected output
         int res2 = g2.hopcroftKarpAlgorithm();
 
         assert(res2 == expected_res2); // assert check to ensure that the algorithm executed correctly for test 2
    
      // Sample test case 3
             int v3a = 6, v3b = 6, e3 = 4;  // vertices of left side, right side and edges
         HKGraph g3(v3a, v3b); // execute the algorithm 
 
             g3.addEdge(0,1);
         g3.addEdge(1,4);
         g3.addEdge(1,5);
         g3.addEdge(5,0);
 
         int expected_res3 = 0; // for the above sample data, this is the expected output
         int res3 = g3.hopcroftKarpAlgorithm();
 
         assert(res3 == expected_res3); // assert check to ensure that the algorithm executed correctly for test 3
    
    
        
}

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