mirror of
https://github.com/TheAlgorithms/C-Plus-Plus.git
synced 2026-04-06 12:09:46 +08:00
265 lines
7.4 KiB
C++
265 lines
7.4 KiB
C++
/**
|
||
* @file
|
||
* @brief Implementation of [Hopcroft–Karp](https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm) algorithm.
|
||
* @details
|
||
* 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](https://github.com/Krishnapal4050)
|
||
*
|
||
*/
|
||
|
||
|
||
#include <iostream>
|
||
#include <cstdlib>
|
||
#include <queue>
|
||
#include <list>
|
||
#include <climits>
|
||
#include <memory>
|
||
|
||
/**
|
||
* @brief Represents Bipartite graph for
|
||
* Hopcroft Karp implementation
|
||
*/
|
||
class BGraph
|
||
{
|
||
// m and n are number of vertices on left
|
||
// and right sides of Bipartite Graph
|
||
int m, n;
|
||
const int NIL;
|
||
const int INF;
|
||
|
||
// adj[u] stores adjacents of left side
|
||
// vertex 'u'. The value of u ranges from 1 to m.
|
||
// 0 is used for dummy vertex
|
||
std::vector<std::list<int> >adj;
|
||
|
||
|
||
//vectors for hopcroftKarp()
|
||
std::vector<int> pair_u;
|
||
std::vector<int> pair_v;
|
||
std::vector<int> dist;
|
||
|
||
public:
|
||
BGraph(); //Default Constructor
|
||
BGraph(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 hopcroftKarpAlgorithm();
|
||
};
|
||
|
||
// Returns size of maximum matching
|
||
int BGraph::hopcroftKarpAlgorithm()
|
||
{
|
||
// pair_u[u] stores pair of u in matching on left side of Bipartite Graph.
|
||
// If u doesn't have any pair, then pair_u[u] is NIL
|
||
pair_u = std::vector<int>(m + 1);
|
||
|
||
// pair_v[v] stores pair of v in matching on right side of Biparite Graph.
|
||
// If v doesn't have any pair, then pair_u[v] is NIL
|
||
pair_v = std::vector<int>(n + 1);
|
||
|
||
// dist[u] stores distance of left side vertices
|
||
dist = std::vector<int>(m + 1);
|
||
|
||
// Initialize NIL as pair of all vertices
|
||
for (int u = 0; u <= m; u++){
|
||
pair_u[u] = NIL;
|
||
}
|
||
for (int v = 0; v <= n; v++){
|
||
pair_v[v] = NIL;
|
||
}
|
||
|
||
// Initialize result
|
||
int result = 0;
|
||
|
||
// Keep updating the result while there is an
|
||
// augmenting path possible.
|
||
while (bfs())
|
||
{
|
||
// Find a free vertex to check for a matching
|
||
for (int u = 1; u <= m; u++){
|
||
|
||
// If current vertex is free and there is
|
||
// an augmenting path from current vertex
|
||
// then increment the result
|
||
if (pair_u[u] == NIL && dfs(u)){
|
||
result++;
|
||
}
|
||
}
|
||
}
|
||
return result;
|
||
}
|
||
|
||
// Returns true if there is an augmenting path available, else returns false
|
||
bool BGraph::bfs()
|
||
{
|
||
std::queue<int> q; //an integer queue for bfs
|
||
|
||
// 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 (pair_u[u] == NIL){
|
||
// u is not matched so distance is 0
|
||
dist[u] = 0;
|
||
q.push(u);
|
||
}
|
||
|
||
// Else set distance as infinite so that this vertex is considered next time for availibility
|
||
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 then
|
||
if (dist[u] < dist[NIL])
|
||
{
|
||
// Get all the adjacent vertices of the dequeued vertex u
|
||
std::list<int>::iterator it;
|
||
for (it = adj[u].begin(); it != adj[u].end(); ++it)
|
||
{
|
||
int v = *it;
|
||
|
||
// If pair of v is not considered so far
|
||
// i.e. (v, pair_v[v]) is not yet explored edge.
|
||
if (dist[pair_v[v]] == INF)
|
||
{
|
||
// Consider the pair and push it to queue
|
||
dist[pair_v[v]] = dist[u] + 1;
|
||
q.push(pair_v[v]);
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
// If we could come back to NIL using alternating path of distinct
|
||
// vertices then there is an augmenting path available
|
||
return (dist[NIL] != INF);
|
||
}
|
||
|
||
// Returns true if there is an augmenting path beginning with free vertex u
|
||
bool BGraph::dfs(int u)
|
||
{
|
||
if (u != NIL)
|
||
{
|
||
std::list<int>::iterator it;
|
||
for (it = adj[u].begin(); it != adj[u].end(); ++it)
|
||
{
|
||
// Adjacent vertex of u
|
||
int v = *it;
|
||
|
||
// Follow the distances set by BFS search
|
||
if (dist[pair_v[v]] == dist[u] + 1)
|
||
{
|
||
// If dfs for pair of v also returnn true then
|
||
if (dfs(pair_v[v]) == true)
|
||
{ // new matching possible, store the matching
|
||
pair_v[v] = u;
|
||
pair_u[u] = v;
|
||
return true;
|
||
}
|
||
}
|
||
}
|
||
|
||
// If there is no augmenting path beginning with u then.
|
||
dist[u] = INF;
|
||
return false;
|
||
}
|
||
return true;
|
||
}
|
||
|
||
// Default Constructor for initialization
|
||
BGraph::BGraph(){
|
||
NIL=0;
|
||
INF=INT_MAX;
|
||
}
|
||
|
||
// Constructor for initialization
|
||
BGraph::BGraph(int m, int n)
|
||
{
|
||
this->BGraph();
|
||
this->m = m;
|
||
this->n = n;
|
||
adj = std::vector<std::list<int> >(m + 1);
|
||
}
|
||
|
||
// function to add edge from u to v
|
||
void BGraph::addEdge(int u, int v)
|
||
{
|
||
adj[u].push_back(v); // Add v to u’s list.
|
||
}
|
||
|
||
|
||
int main()
|
||
{
|
||
int v1 = 0, v2 = 0, e = 0;
|
||
std::cin >> v1 >> v2 >> e; // vertices of left side, right side and edges
|
||
BGraph 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;
|
||
|
||
}
|