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Add Edmonds Blossom Algorithm (#5471)
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Tarun Vishwakarma
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package com.thealgorithms.datastructures.graphs;
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import java.util.ArrayList;
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import java.util.Arrays;
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import java.util.LinkedList;
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import java.util.List;
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import java.util.Queue;
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/**
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* The EdmondsBlossomAlgorithm class implements Edmonds' Blossom Algorithm
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* to find the maximum matching in a general graph. The algorithm efficiently
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* handles cases where the graph contains odd-length cycles by contracting
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* "blossoms" and finding augmenting paths.
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*<p>
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* <a href="https://stanford.edu/~rezab/classes/cme323/S16/projects_reports/shoemaker_vare.pdf">Documentation of Algorithm (Stanford University)</a>
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* <p></p>
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* <a href="https://en.wikipedia.org/wiki/Blossom_algorithm">Wikipedia Documentation</a>
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*/
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public final class EdmondsBlossomAlgorithm {
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private EdmondsBlossomAlgorithm() {
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}
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private static final int UNMATCHED = -1; // Constant to represent unmatched vertices
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/**
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* Finds the maximum matching in a general graph (Edmonds Blossom Algorithm).
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*
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* @param edges A list of edges in the graph.
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* @param vertexCount The number of vertices in the graph.
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* @return A list of matched pairs of vertices.
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*/
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public static List<int[]> maximumMatching(List<int[]> edges, int vertexCount) {
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List<List<Integer>> graph = new ArrayList<>(vertexCount);
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// Initialize each vertex's adjacency list.
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for (int i = 0; i < vertexCount; i++) {
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graph.add(new ArrayList<>());
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}
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// Populate the graph with the edges
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for (int[] edge : edges) {
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int u = edge[0];
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int v = edge[1];
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graph.get(u).add(v);
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graph.get(v).add(u);
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}
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// Initial matching array and auxiliary data structures
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int[] match = new int[vertexCount];
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Arrays.fill(match, UNMATCHED); // All vertices are initially unmatched
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int[] parent = new int[vertexCount];
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int[] base = new int[vertexCount];
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boolean[] inBlossom = new boolean[vertexCount]; // Indicates if a vertex is part of a blossom
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boolean[] inQueue = new boolean[vertexCount]; // Tracks vertices in the BFS queue
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// Main logic for finding maximum matching
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for (int u = 0; u < vertexCount; u++) {
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if (match[u] == UNMATCHED) {
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// BFS initialization
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Arrays.fill(parent, UNMATCHED);
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for (int i = 0; i < vertexCount; i++) {
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base[i] = i; // Each vertex is its own base initially
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}
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Arrays.fill(inBlossom, false);
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Arrays.fill(inQueue, false);
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Queue<Integer> queue = new LinkedList<>();
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queue.add(u);
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inQueue[u] = true;
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boolean augmentingPathFound = false;
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// BFS to find augmenting paths
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while (!queue.isEmpty() && !augmentingPathFound) {
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int current = queue.poll(); // Use a different name for clarity
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for (int y : graph.get(current)) {
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// Skip if we are looking at the same edge as the current match
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if (match[current] == y) {
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continue;
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}
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if (base[current] == base[y]) {
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continue; // Avoid self-loops
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}
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if (parent[y] == UNMATCHED) {
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// Case 1: y is unmatched, we've found an augmenting path
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if (match[y] == UNMATCHED) {
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parent[y] = current;
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augmentingPathFound = true;
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updateMatching(match, parent, y); // Augment along this path
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break;
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}
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// Case 2: y is matched, add y's match to the queue
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int z = match[y];
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parent[y] = current;
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parent[z] = y;
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if (!inQueue[z]) {
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queue.add(z);
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inQueue[z] = true;
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}
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} else {
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// Case 3: Both x and y have a parent; check for a cycle/blossom
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int baseU = findBase(base, parent, current, y);
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if (baseU != UNMATCHED) {
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contractBlossom(new BlossomData(new BlossomAuxData(queue, parent, base, inBlossom, match, inQueue), current, y, baseU));
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}
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}
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}
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}
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}
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}
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// Create result list of matched pairs
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List<int[]> matchingResult = new ArrayList<>();
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for (int v = 0; v < vertexCount; v++) {
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if (match[v] != UNMATCHED && v < match[v]) {
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matchingResult.add(new int[] {v, match[v]});
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}
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}
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return matchingResult;
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}
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/**
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* Updates the matching along the augmenting path found.
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*
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* @param match The matching array.
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* @param parent The parent array used during the BFS.
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* @param u The starting node of the augmenting path.
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*/
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private static void updateMatching(int[] match, int[] parent, int u) {
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while (u != UNMATCHED) {
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int v = parent[u];
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int next = match[v];
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match[v] = u;
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match[u] = v;
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u = next;
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}
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}
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/**
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* Finds the base of a node in the blossom.
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*
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* @param base The base array.
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* @param parent The parent array.
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* @param u One end of the edge.
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* @param v The other end of the edge.
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* @return The base of the node or UNMATCHED.
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*/
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private static int findBase(int[] base, int[] parent, int u, int v) {
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boolean[] visited = new boolean[base.length];
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// Mark ancestors of u
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int currentU = u;
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while (true) {
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currentU = base[currentU]; // Move assignment out of the condition
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visited[currentU] = true;
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if (parent[currentU] == UNMATCHED) {
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break;
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}
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currentU = parent[currentU]; // Move assignment out of the condition
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}
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// Find the common ancestor of v
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int currentV = v;
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while (true) {
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currentV = base[currentV]; // Move assignment out of the condition
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if (visited[currentV]) {
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return currentV;
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}
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currentV = parent[currentV]; // Move assignment out of the condition
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}
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}
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/**
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* Contracts a blossom and updates the base array.
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*
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* @param blossomData The data containing the parameters related to the blossom contraction.
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*/
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private static void contractBlossom(BlossomData blossomData) {
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for (int x = blossomData.u; blossomData.auxData.base[x] != blossomData.lca; x = blossomData.auxData.parent[blossomData.auxData.match[x]]) {
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int baseX = blossomData.auxData.base[x];
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int matchBaseX = blossomData.auxData.base[blossomData.auxData.match[x]];
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// Split the inner assignment into two separate assignments
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blossomData.auxData.inBlossom[baseX] = true;
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blossomData.auxData.inBlossom[matchBaseX] = true;
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}
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for (int x = blossomData.v; blossomData.auxData.base[x] != blossomData.lca; x = blossomData.auxData.parent[blossomData.auxData.match[x]]) {
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int baseX = blossomData.auxData.base[x];
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int matchBaseX = blossomData.auxData.base[blossomData.auxData.match[x]];
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// Split the inner assignment into two separate assignments
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blossomData.auxData.inBlossom[baseX] = true;
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blossomData.auxData.inBlossom[matchBaseX] = true;
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}
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// Update the base for all marked vertices
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for (int i = 0; i < blossomData.auxData.base.length; i++) {
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if (blossomData.auxData.inBlossom[blossomData.auxData.base[i]]) {
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blossomData.auxData.base[i] = blossomData.lca; // Contract to the lowest common ancestor
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if (!blossomData.auxData.inQueue[i]) {
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blossomData.auxData.queue.add(i); // Add to queue if not already present
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blossomData.auxData.inQueue[i] = true;
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}
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}
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}
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}
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/**
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* Auxiliary data class to encapsulate common parameters for the blossom operations.
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*/
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static class BlossomAuxData {
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Queue<Integer> queue; // Queue for BFS traversal
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int[] parent; // Parent array to store the paths
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int[] base; // Base array to track the base of each vertex
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boolean[] inBlossom; // Flags to indicate if a vertex is in a blossom
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int[] match; // Array to store matches for each vertex
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boolean[] inQueue; // Flags to track vertices in the BFS queue
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BlossomAuxData(Queue<Integer> queue, int[] parent, int[] base, boolean[] inBlossom, int[] match, boolean[] inQueue) {
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this.queue = queue;
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this.parent = parent;
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this.base = base;
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this.inBlossom = inBlossom;
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this.match = match;
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this.inQueue = inQueue;
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}
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}
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/**
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* BlossomData class with reduced parameters.
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*/
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static class BlossomData {
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BlossomAuxData auxData; // Use the auxiliary data class
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int u; // One vertex in the edge
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int v; // Another vertex in the edge
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int lca; // Lowest Common Ancestor
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BlossomData(BlossomAuxData auxData, int u, int v, int lca) {
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this.auxData = auxData;
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this.u = u;
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this.v = v;
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this.lca = lca;
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}
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}
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}
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