Below is the syntax highlighted version of BipartiteMatching.java
from §6.5 Reductions.
/****************************************************************************** * Compilation: javac BipartiteMatching.java * Execution: java BipartiteMatching V1 V2 E * Dependencies: BipartiteX.java * * Find a maximum cardinality matching (and minimum cardinality vertex cover) * in a bipartite graph using the alternating path algorithm. * ******************************************************************************/ /** * The {@code BipartiteMatching} 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 bipartite * 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>alternating-path algorithm</em>. * It is equivalent to reducing to the maximum-flow problem and running * the augmenting-path algorithm on the resulting flow network, but it * does so with less overhead. * The constructor takes <em>O</em>((<em>E</em> + <em>V</em>) <em>V</em>) * time, where <em>E</em> is the number of edges and <em>V</em> is the * number of vertices in the graph. * It uses Θ(<em>V</em>) extra space (not including the graph). * <p> * See also {@link HopcroftKarp}, which solves the problem in * <em>O</em>(<em>E</em> sqrt(<em>V</em>)) using the Hopcroft-Karp * algorithm and * <a href = "https://algs4.cs.princeton.edu/65reductions/BipartiteMatchingToMaxflow.java.html">BipartiteMatchingToMaxflow</a>, * which solves the problem in <em>O</em>((<em>E</em> + <em>V</em>) <em>V</em>) * time via a reduction to maxflow. * <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 BipartiteMatching { 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[] edgeTo; // edgeTo[v] = last edge on alternating 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 BipartiteMatching(Graph G) { bipartition = new BipartiteX(G); if (!bipartition.isBipartite()) { throw new IllegalArgumentException("graph is not bipartite"); } this.V = G.V(); // initialize empty matching mate = new int[V]; for (int v = 0; v < V; v++) mate[v] = UNMATCHED; // alternating path algorithm while (hasAugmentingPath(G)) { // find one endpoint t in alternating path int t = -1; for (int v = 0; v < G.V(); v++) { if (!isMatched(v) && edgeTo[v] != -1) { t = v; break; } } // update the matching according to alternating path in edgeTo[] array for (int v = t; v != -1; v = edgeTo[edgeTo[v]]) { int w = edgeTo[v]; mate[v] = w; mate[w] = v; } cardinality++; } // find min vertex cover from marked[] array 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); } /* * 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 * * this implementation finds a shortest augmenting path (fewest number of edges), though there * is no particular advantage to do so here */ private boolean hasAugmentingPath(Graph G) { marked = new boolean[V]; edgeTo = new int[V]; for (int v = 0; v < V; v++) edgeTo[v] = -1; // 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; } } // run BFS, stopping as soon as an alternating path is found while (!queue.isEmpty()) { int v = queue.dequeue(); for (int w : G.adj(v)) { // either (1) forward edge not in matching or (2) backward edge in matching if (isResidualGraphEdge(v, w) && !marked[w]) { edgeTo[w] = v; marked[w] = true; if (!isMatched(w)) return true; queue.enqueue(w); } } } return false; } // 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; } /** * 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 a maximum matching. * * @return the number of edges in a 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]; } 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[0]); int V2 = Integer.parseInt(args[1]); int E = Integer.parseInt(args[2]); Graph G = GraphGenerator.bipartite(V1, V2, E); if (G.V() < 1000) StdOut.println(G); BipartiteMatching matching = new BipartiteMatching(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(); } }