# HopcroftKarp.java

Below is the syntax highlighted version of HopcroftKarp.java.

```/******************************************************************************
*  Compilation:  javac HopcroftKarp.java
*  Execution:    java HopcroftKarp V1 V2 E
*  Dependencies: FordFulkerson.java FlowNetwork.java FlowEdge.java
*                BipartiteX.java
*
*  Find a maximum cardinality matching (and minimum cardinality vertex cover)
*  in a bipartite graph using Hopcroft-Karp algorithm.
*
******************************************************************************/

package edu.princeton.cs.algs4;

import java.util.Iterator;

/**
*  The {@code HopcroftKarp} class represents a data type for computing a
*  <em>maximum (cardinality) matching</em> and a
*  <em>minimum (cardinality) vertex cover</em> in a bipartite graph.
*  A <em>bipartite graph</em> in a graph whose vertices can be partitioned
*  into two disjoint sets such that every edge has one endpoint in either set.
*  A <em>matching</em> in a graph is a subset of its edges with no common
*  vertices. A <em>maximum matching</em> is a matching with the maximum number
*  of edges.
*  A <em>perfect matching</em> is a matching which matches all vertices in the graph.
*  A <em>vertex cover</em> in a graph is a subset of its vertices such that
*  every edge is incident to at least one vertex. A <em>minimum vertex cover</em>
*  is a vertex cover with the minimum number of vertices.
*  By Konig's theorem, in any biparite
*  graph, the maximum number of edges in matching equals the minimum number
*  of vertices in a vertex cover.
*  The maximum matching problem in <em>nonbipartite</em> graphs is
*  also important, but all known algorithms for this more general problem
*  are substantially more complicated.
*  <p>
*  This implementation uses the <em>Hopcroft-Karp algorithm</em>.
*  The order of growth of the running time in the worst case is
*  (<em>E</em> + <em>V</em>) sqrt(<em>V</em>),
*  where <em>E</em> is the number of edges and <em>V</em> is the number
*  of vertices in the graph. It uses extra space (not including the graph)
*  proportional to <em>V</em>.
*  <p>
*  See also {@link BipartiteMatching}, which solves the problem in
*  O(<em>E V</em>) time using the <em>alternating path algorithm</em>
*  and <a href = "https://algs4.cs.princeton.edu/65reductions/BipartiteMatchingToMaxflow.java.html">BipartiteMatchingToMaxflow</a>,
*  which solves the problem in O(<em>E V</em>) time via a reduction
*  to the maxflow problem.
*  <p>
*  For additional documentation, see
*  <a href="https://algs4.cs.princeton.edu/65reductions">Section 6.5</a>
*  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*
*  @author Robert Sedgewick
*  @author Kevin Wayne
*/
public class HopcroftKarp {
private static final int UNMATCHED = -1;

private final int V;                 // number of vertices in the graph
private BipartiteX bipartition;      // the bipartition
private int cardinality;             // cardinality of current matching
private int[] mate;                  // mate[v] =  w if v-w is an edge in current matching
//         = -1 if v is not in current matching
private boolean[] inMinVertexCover;  // inMinVertexCover[v] = true iff v is in min vertex cover
private boolean[] marked;            // marked[v] = true iff v is reachable via alternating path
private int[] distTo;                // distTo[v] = number of edges on shortest path to v

/**
* Determines a maximum matching (and a minimum vertex cover)
* in a bipartite graph.
*
* @param  G the bipartite graph
* @throws IllegalArgumentException if {@code G} is not bipartite
*/
public HopcroftKarp(Graph G) {
bipartition = new BipartiteX(G);
if (!bipartition.isBipartite()) {
throw new IllegalArgumentException("graph is not bipartite");
}

// initialize empty matching
this.V = G.V();
mate = new int[V];
for (int v = 0; v < V; v++)
mate[v] = UNMATCHED;

// the call to hasAugmentingPath() provides enough info to reconstruct level graph
while (hasAugmentingPath(G)) {

// to be able to iterate over each adjacency list, keeping track of which
// vertex in each adjacency list needs to be explored next
Iterator<Integer>[] adj = (Iterator<Integer>[]) new Iterator[G.V()];
for (int v = 0; v < G.V(); v++)
adj[v] = G.adj(v).iterator();

// for each unmatched vertex s on one side of bipartition
for (int s = 0; s < V; s++) {
if (isMatched(s) || !bipartition.color(s)) continue;   // or use distTo[s] == 0

// find augmenting path from s using nonrecursive DFS
Stack<Integer> path = new Stack<Integer>();
path.push(s);
while (!path.isEmpty()) {
int v = path.peek();

// retreat, no more edges in level graph leaving v
if (!adj[v].hasNext())
path.pop();

// advance
else {
// process edge v-w only if it is an edge in level graph
int w = adj[v].next();
if (!isLevelGraphEdge(v, w)) continue;

// add w to augmenting path
path.push(w);

// augmenting path found: update the matching
if (!isMatched(w)) {
// StdOut.println("augmenting path: " + toString(path));

while (!path.isEmpty()) {
int x = path.pop();
int y = path.pop();
mate[x] = y;
mate[y] = x;
}
cardinality++;
}
}
}
}
}

// also find a min vertex cover
inMinVertexCover = new boolean[V];
for (int v = 0; v < V; v++) {
if (bipartition.color(v) && !marked[v]) inMinVertexCover[v] = true;
if (!bipartition.color(v) && marked[v]) inMinVertexCover[v] = true;
}

assert certifySolution(G);
}

// string representation of augmenting path (chop off last vertex)
private static String toString(Iterable<Integer> path) {
StringBuilder sb = new StringBuilder();
for (int v : path)
sb.append(v + "-");
String s = sb.toString();
s = s.substring(0, s.lastIndexOf('-'));
return s;
}

// is the edge v-w in the level graph?
private boolean isLevelGraphEdge(int v, int w) {
return (distTo[w] == distTo[v] + 1) && isResidualGraphEdge(v, w);
}

// is the edge v-w a forward edge not in the matching or a reverse edge in the matching?
private boolean isResidualGraphEdge(int v, int w) {
if ((mate[v] != w) &&  bipartition.color(v)) return true;
if ((mate[v] == w) && !bipartition.color(v)) return true;
return false;
}

/*
* is there an augmenting path?
*   - if so, upon termination adj[] contains the level graph;
*   - if not, upon termination marked[] specifies those vertices reachable via an alternating
*     path from one side of the bipartition
*
* an alternating path is a path whose edges belong alternately to the matching and not
* to the matching
*
* an augmenting path is an alternating path that starts and ends at unmatched vertices
*/
private boolean hasAugmentingPath(Graph G) {

// shortest path distances
marked = new boolean[V];
distTo = new int[V];
for (int v = 0; v < V; v++)
distTo[v] = Integer.MAX_VALUE;

// breadth-first search (starting from all unmatched vertices on one side of bipartition)
Queue<Integer> queue = new Queue<Integer>();
for (int v = 0; v < V; v++) {
if (bipartition.color(v) && !isMatched(v)) {
queue.enqueue(v);
marked[v] = true;
distTo[v] = 0;
}
}

// run BFS until an augmenting path is found
// (and keep going until all vertices at that distance are explored)
boolean hasAugmentingPath = false;
while (!queue.isEmpty()) {
int v = queue.dequeue();
for (int w : G.adj(v)) {

// forward edge not in matching or backwards edge in matching
if (isResidualGraphEdge(v, w)) {
if (!marked[w]) {
distTo[w] = distTo[v] + 1;
marked[w] = true;
if (!isMatched(w))
hasAugmentingPath = true;

// stop enqueuing vertices once an alternating path has been discovered
// (no vertex on same side will be marked if its shortest path distance longer)
if (!hasAugmentingPath) queue.enqueue(w);
}
}
}
}

return hasAugmentingPath;
}

/**
* Returns the vertex to which the specified vertex is matched in
* the maximum matching computed by the algorithm.
*
* @param  v the vertex
* @return the vertex to which vertex {@code v} is matched in the
*         maximum matching; {@code -1} if the vertex is not matched
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*
*/
public int mate(int v) {
validate(v);
return mate[v];
}

/**
* Returns true if the specified vertex is matched in the maximum matching
* computed by the algorithm.
*
* @param  v the vertex
* @return {@code true} if vertex {@code v} is matched in maximum matching;
*         {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*
*/
public boolean isMatched(int v) {
validate(v);
return mate[v] != UNMATCHED;
}

/**
* Returns the number of edges in any maximum matching.
*
* @return the number of edges in any maximum matching
*/
public int size() {
return cardinality;
}

/**
* Returns true if the graph contains a perfect matching.
* That is, the number of edges in a maximum matching is equal to one half
* of the number of vertices in the graph (so that every vertex is matched).
*
* @return {@code true} if the graph contains a perfect matching;
*         {@code false} otherwise
*/
public boolean isPerfect() {
return cardinality * 2 == V;
}

/**
* Returns true if the specified vertex is in the minimum vertex cover
* computed by the algorithm.
*
* @param  v the vertex
* @return {@code true} if vertex {@code v} is in the minimum vertex cover;
*         {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public boolean inMinVertexCover(int v) {
validate(v);
return inMinVertexCover[v];
}

// throw an exception if vertex is invalid
private void validate(int v) {
if (v < 0 || v >= V)
throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
}

/**************************************************************************
*
*  The code below is solely for testing correctness of the data type.
*
**************************************************************************/

// check that mate[] and inVertexCover[] define a max matching and min vertex cover, respectively
private boolean certifySolution(Graph G) {

// check that mate(v) = w iff mate(w) = v
for (int v = 0; v < V; v++) {
if (mate(v) == -1) continue;
if (mate(mate(v)) != v) return false;
}

// check that size() is consistent with mate()
int matchedVertices = 0;
for (int v = 0; v < V; v++) {
if (mate(v) != -1) matchedVertices++;
}
if (2*size() != matchedVertices) return false;

// check that size() is consistent with minVertexCover()
int sizeOfMinVertexCover = 0;
for (int v = 0; v < V; v++)
if (inMinVertexCover(v)) sizeOfMinVertexCover++;
if (size() != sizeOfMinVertexCover) return false;

// check that mate() uses each vertex at most once
boolean[] isMatched = new boolean[V];
for (int v = 0; v < V; v++) {
int w = mate[v];
if (w == -1) continue;
if (v == w) return false;
if (v >= w) continue;
if (isMatched[v] || isMatched[w]) return false;
isMatched[v] = true;
isMatched[w] = true;
}

// check that mate() uses only edges that appear in the graph
for (int v = 0; v < V; v++) {
if (mate(v) == -1) continue;
boolean isEdge = false;
for (int w : G.adj(v)) {
if (mate(v) == w) isEdge = true;
}
if (!isEdge) return false;
}

// check that inMinVertexCover() is a vertex cover
for (int v = 0; v < V; v++)
for (int w : G.adj(v))
if (!inMinVertexCover(v) && !inMinVertexCover(w)) return false;

return true;
}

/**
* Unit tests the {@code HopcroftKarp} data type.
* Takes three command-line arguments {@code V1}, {@code V2}, and {@code E};
* creates a random bipartite graph with {@code V1} + {@code V2} vertices
* and {@code E} edges; computes a maximum matching and minimum vertex cover;
* and prints the results.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {

int V1 = Integer.parseInt(args);
int V2 = Integer.parseInt(args);
int E  = Integer.parseInt(args);
Graph G = GraphGenerator.bipartite(V1, V2, E);
if (G.V() < 1000) StdOut.println(G);

HopcroftKarp matching = new HopcroftKarp(G);

// print maximum matching
StdOut.printf("Number of edges in max matching        = %d\n", matching.size());
StdOut.printf("Number of vertices in min vertex cover = %d\n", matching.size());
StdOut.printf("Graph has a perfect matching           = %b\n", matching.isPerfect());
StdOut.println();

if (G.V() >= 1000) return;

StdOut.print("Max matching: ");
for (int v = 0; v < G.V(); v++) {
int w = matching.mate(v);
if (matching.isMatched(v) && v < w)  // print each edge only once
StdOut.print(v + "-" + w + " ");
}
StdOut.println();

// print minimum vertex cover
StdOut.print("Min vertex cover: ");
for (int v = 0; v < G.V(); v++)
if (matching.inMinVertexCover(v))
StdOut.print(v + " ");
StdOut.println();
}

}

/******************************************************************************
*  Copyright 2002-2020, Robert Sedgewick and Kevin Wayne.
*
*  This file is part of algs4.jar, which accompanies the textbook
*
*      Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
*      Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
*      http://algs4.cs.princeton.edu
*
*
*  algs4.jar is free software: you can redistribute it and/or modify
*  it under the terms of the GNU General Public License as published by
*  the Free Software Foundation, either version 3 of the License, or
*  (at your option) any later version.
*
*  algs4.jar is distributed in the hope that it will be useful,
*  but WITHOUT ANY WARRANTY; without even the implied warranty of
*  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
*  GNU General Public License for more details.
*
*  You should have received a copy of the GNU General Public License
*  along with algs4.jar.  If not, see http://www.gnu.org/licenses.
******************************************************************************/
```

Last updated: Wed Jun 9 05:11:22 EDT 2021.