/*
* 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;
}
/**
* Unit tests the {@code BipartiteX} data type.
*
* @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]);
int F = Integer.parseInt(args[3]);
// create random bipartite graph with V1 vertices on left side,
// V2 vertices on right side, and E edges; then add F random edges
Graph G = GraphGenerator.bipartite(V1, V2, E);
for (int i = 0; i < F; i++) {
int v = StdRandom.uniform(V1 + V2);
int w = StdRandom.uniform(V1 + V2);
G.addEdge(v, w);
}
StdOut.println(G);
BipartiteX b = new BipartiteX(G);
if (b.isBipartite()) {
StdOut.println("Graph is bipartite");
for (int v = 0; v < G.V(); v++) {
StdOut.println(v + ": " + b.color(v));
}
}
else {
StdOut.print("Graph has an odd-length cycle: ");
for (int x : b.oddCycle()) {
StdOut.print(x + " ");
}
StdOut.println();
}
}
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);
}
