diff --git a/DIRECTORY.md b/DIRECTORY.md index ee09790ed64d..6f63a88b085a 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -107,6 +107,7 @@ * [ConnectedComponent](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/ConnectedComponent.java) * [Cycles](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/Cycles.java) * [DijkstraAlgorithm](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/DijkstraAlgorithm.java) + * [EdmondsBlossomAlgorithm](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithm.java) * [FloydWarshall](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/FloydWarshall.java) * [FordFulkerson](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/FordFulkerson.java) * [Graphs](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/datastructures/graphs/Graphs.java) @@ -659,6 +660,7 @@ * graphs * [BoruvkaAlgorithmTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/BoruvkaAlgorithmTest.java) * [DijkstraAlgorithmTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/DijkstraAlgorithmTest.java) + * [EdmondsBlossomAlgorithmTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithmTest.java) * [FordFulkersonTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/FordFulkersonTest.java) * [HamiltonianCycleTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/HamiltonianCycleTest.java) * [KosarajuTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/datastructures/graphs/KosarajuTest.java) diff --git a/src/main/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithm.java b/src/main/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithm.java new file mode 100644 index 000000000000..27ad96d71876 --- /dev/null +++ b/src/main/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithm.java @@ -0,0 +1,251 @@ +package com.thealgorithms.datastructures.graphs; + +import java.util.ArrayList; +import java.util.Arrays; +import java.util.LinkedList; +import java.util.List; +import java.util.Queue; + +/** + * The EdmondsBlossomAlgorithm class implements Edmonds' Blossom Algorithm + * to find the maximum matching in a general graph. The algorithm efficiently + * handles cases where the graph contains odd-length cycles by contracting + * "blossoms" and finding augmenting paths. + *

+ * Documentation of Algorithm (Stanford University) + *

+ * Wikipedia Documentation + */ +public final class EdmondsBlossomAlgorithm { + + private EdmondsBlossomAlgorithm() { + } + + private static final int UNMATCHED = -1; // Constant to represent unmatched vertices + + /** + * Finds the maximum matching in a general graph (Edmonds Blossom Algorithm). + * + * @param edges A list of edges in the graph. + * @param vertexCount The number of vertices in the graph. + * @return A list of matched pairs of vertices. + */ + public static List maximumMatching(List edges, int vertexCount) { + List> graph = new ArrayList<>(vertexCount); + + // Initialize each vertex's adjacency list. + for (int i = 0; i < vertexCount; i++) { + graph.add(new ArrayList<>()); + } + + // Populate the graph with the edges + for (int[] edge : edges) { + int u = edge[0]; + int v = edge[1]; + graph.get(u).add(v); + graph.get(v).add(u); + } + + // Initial matching array and auxiliary data structures + int[] match = new int[vertexCount]; + Arrays.fill(match, UNMATCHED); // All vertices are initially unmatched + int[] parent = new int[vertexCount]; + int[] base = new int[vertexCount]; + boolean[] inBlossom = new boolean[vertexCount]; // Indicates if a vertex is part of a blossom + boolean[] inQueue = new boolean[vertexCount]; // Tracks vertices in the BFS queue + + // Main logic for finding maximum matching + for (int u = 0; u < vertexCount; u++) { + if (match[u] == UNMATCHED) { + // BFS initialization + Arrays.fill(parent, UNMATCHED); + for (int i = 0; i < vertexCount; i++) { + base[i] = i; // Each vertex is its own base initially + } + Arrays.fill(inBlossom, false); + Arrays.fill(inQueue, false); + + Queue queue = new LinkedList<>(); + queue.add(u); + inQueue[u] = true; + + boolean augmentingPathFound = false; + + // BFS to find augmenting paths + while (!queue.isEmpty() && !augmentingPathFound) { + int current = queue.poll(); // Use a different name for clarity + for (int y : graph.get(current)) { + // Skip if we are looking at the same edge as the current match + if (match[current] == y) { + continue; + } + + if (base[current] == base[y]) { + continue; // Avoid self-loops + } + + if (parent[y] == UNMATCHED) { + // Case 1: y is unmatched, we've found an augmenting path + if (match[y] == UNMATCHED) { + parent[y] = current; + augmentingPathFound = true; + updateMatching(match, parent, y); // Augment along this path + break; + } + + // Case 2: y is matched, add y's match to the queue + int z = match[y]; + parent[y] = current; + parent[z] = y; + if (!inQueue[z]) { + queue.add(z); + inQueue[z] = true; + } + } else { + // Case 3: Both x and y have a parent; check for a cycle/blossom + int baseU = findBase(base, parent, current, y); + if (baseU != UNMATCHED) { + contractBlossom(new BlossomData(new BlossomAuxData(queue, parent, base, inBlossom, match, inQueue), current, y, baseU)); + } + } + } + } + } + } + + // Create result list of matched pairs + List matchingResult = new ArrayList<>(); + for (int v = 0; v < vertexCount; v++) { + if (match[v] != UNMATCHED && v < match[v]) { + matchingResult.add(new int[] {v, match[v]}); + } + } + + return matchingResult; + } + + /** + * Updates the matching along the augmenting path found. + * + * @param match The matching array. + * @param parent The parent array used during the BFS. + * @param u The starting node of the augmenting path. + */ + private static void updateMatching(int[] match, int[] parent, int u) { + while (u != UNMATCHED) { + int v = parent[u]; + int next = match[v]; + match[v] = u; + match[u] = v; + u = next; + } + } + + /** + * Finds the base of a node in the blossom. + * + * @param base The base array. + * @param parent The parent array. + * @param u One end of the edge. + * @param v The other end of the edge. + * @return The base of the node or UNMATCHED. + */ + private static int findBase(int[] base, int[] parent, int u, int v) { + boolean[] visited = new boolean[base.length]; + + // Mark ancestors of u + int currentU = u; + while (true) { + currentU = base[currentU]; // Move assignment out of the condition + visited[currentU] = true; + if (parent[currentU] == UNMATCHED) { + break; + } + currentU = parent[currentU]; // Move assignment out of the condition + } + + // Find the common ancestor of v + int currentV = v; + while (true) { + currentV = base[currentV]; // Move assignment out of the condition + if (visited[currentV]) { + return currentV; + } + currentV = parent[currentV]; // Move assignment out of the condition + } + } + + /** + * Contracts a blossom and updates the base array. + * + * @param blossomData The data containing the parameters related to the blossom contraction. + */ + private static void contractBlossom(BlossomData blossomData) { + for (int x = blossomData.u; blossomData.auxData.base[x] != blossomData.lca; x = blossomData.auxData.parent[blossomData.auxData.match[x]]) { + int baseX = blossomData.auxData.base[x]; + int matchBaseX = blossomData.auxData.base[blossomData.auxData.match[x]]; + + // Split the inner assignment into two separate assignments + blossomData.auxData.inBlossom[baseX] = true; + blossomData.auxData.inBlossom[matchBaseX] = true; + } + + for (int x = blossomData.v; blossomData.auxData.base[x] != blossomData.lca; x = blossomData.auxData.parent[blossomData.auxData.match[x]]) { + int baseX = blossomData.auxData.base[x]; + int matchBaseX = blossomData.auxData.base[blossomData.auxData.match[x]]; + + // Split the inner assignment into two separate assignments + blossomData.auxData.inBlossom[baseX] = true; + blossomData.auxData.inBlossom[matchBaseX] = true; + } + + // Update the base for all marked vertices + for (int i = 0; i < blossomData.auxData.base.length; i++) { + if (blossomData.auxData.inBlossom[blossomData.auxData.base[i]]) { + blossomData.auxData.base[i] = blossomData.lca; // Contract to the lowest common ancestor + if (!blossomData.auxData.inQueue[i]) { + blossomData.auxData.queue.add(i); // Add to queue if not already present + blossomData.auxData.inQueue[i] = true; + } + } + } + } + + /** + * Auxiliary data class to encapsulate common parameters for the blossom operations. + */ + static class BlossomAuxData { + Queue queue; // Queue for BFS traversal + int[] parent; // Parent array to store the paths + int[] base; // Base array to track the base of each vertex + boolean[] inBlossom; // Flags to indicate if a vertex is in a blossom + int[] match; // Array to store matches for each vertex + boolean[] inQueue; // Flags to track vertices in the BFS queue + + BlossomAuxData(Queue queue, int[] parent, int[] base, boolean[] inBlossom, int[] match, boolean[] inQueue) { + this.queue = queue; + this.parent = parent; + this.base = base; + this.inBlossom = inBlossom; + this.match = match; + this.inQueue = inQueue; + } + } + + /** + * BlossomData class with reduced parameters. + */ + static class BlossomData { + BlossomAuxData auxData; // Use the auxiliary data class + int u; // One vertex in the edge + int v; // Another vertex in the edge + int lca; // Lowest Common Ancestor + + BlossomData(BlossomAuxData auxData, int u, int v, int lca) { + this.auxData = auxData; + this.u = u; + this.v = v; + this.lca = lca; + } + } +} diff --git a/src/test/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithmTest.java b/src/test/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithmTest.java new file mode 100644 index 000000000000..4a7232447e50 --- /dev/null +++ b/src/test/java/com/thealgorithms/datastructures/graphs/EdmondsBlossomAlgorithmTest.java @@ -0,0 +1,119 @@ +package com.thealgorithms.datastructures.graphs; + +import static org.junit.jupiter.api.Assertions.assertArrayEquals; +import static org.junit.jupiter.api.Assertions.assertEquals; +import static org.junit.jupiter.api.Assertions.assertTrue; + +import java.util.Arrays; +import java.util.Collections; +import java.util.List; +import org.junit.jupiter.api.Test; + +/** + * Unit tests for the EdmondsBlossomAlgorithm class. + * + * These tests ensure that the Edmonds' Blossom Algorithm implementation + * works as expected for various graph structures, returning the correct + * maximum matching. + */ +public class EdmondsBlossomAlgorithmTest { + + /** + * Helper method to convert a list of matching pairs into a sorted 2D array. + * Sorting ensures consistent ordering of pairs and vertices for easier comparison in tests. + * + * @param matching List of matched pairs returned by the algorithm. + * @return A sorted 2D array of matching pairs. + */ + private int[][] convertMatchingToArray(List matching) { + // Convert the list of pairs into an array + int[][] result = matching.toArray(new int[0][]); + + // Sort each individual pair for consistency + for (int[] pair : result) { + Arrays.sort(pair); + } + + // Sort the array of pairs to ensure consistent order + Arrays.sort(result, (a, b) -> Integer.compare(a[0], b[0])); + return result; + } + + /** + * Test Case 1: A triangle graph where vertices 0, 1, and 2 form a cycle. + * The expected maximum matching is a single pair (0, 1) or any equivalent pair from the cycle. + */ + @Test + public void testCase1() { + List edges = Arrays.asList(new int[] {0, 1}, new int[] {1, 2}, new int[] {2, 0}); + List matching = EdmondsBlossomAlgorithm.maximumMatching(edges, 3); + + int[][] expected = new int[][] {{0, 1}}; + assertArrayEquals(expected, convertMatchingToArray(matching)); + } + + /** + * Test Case 2: A disconnected graph where vertices 0, 1, 2 form one component, + * and vertices 3, 4 form another. The expected maximum matching is two pairs: + * (0, 1) and (3, 4). + */ + @Test + public void testCase2() { + List edges = Arrays.asList(new int[] {0, 1}, new int[] {1, 2}, new int[] {3, 4}); + List matching = EdmondsBlossomAlgorithm.maximumMatching(edges, 5); + + int[][] expected = new int[][] {{0, 1}, {3, 4}}; + assertArrayEquals(expected, convertMatchingToArray(matching)); + } + + /** + * Test Case 3: A cycle graph involving vertices 0, 1, 2, 3 forming a cycle, + * with an additional edge (4, 5) outside the cycle. + * The expected maximum matching is (0, 1) and (4, 5). + */ + @Test + public void testCase3() { + List edges = Arrays.asList(new int[] {0, 1}, new int[] {1, 2}, new int[] {2, 3}, new int[] {3, 0}, new int[] {4, 5}); + List matching = EdmondsBlossomAlgorithm.maximumMatching(edges, 6); + + // Updated expected output to include the maximum matching pairs + int[][] expected = new int[][] {{0, 1}, {2, 3}, {4, 5}}; + assertArrayEquals(expected, convertMatchingToArray(matching)); + } + + /** + * Test Case 4: A graph with no edges. + * Since there are no edges, the expected matching is an empty set. + */ + @Test + public void testCaseNoMatching() { + List edges = Collections.emptyList(); // No edges + List matching = EdmondsBlossomAlgorithm.maximumMatching(edges, 3); + + int[][] expected = new int[][] {}; // No pairs expected + assertArrayEquals(expected, convertMatchingToArray(matching)); + } + + /** + * Test Case 5: A more complex graph with multiple cycles and extra edges. + * This tests the algorithm's ability to handle larger, more intricate graphs. + * The expected matching is {{0, 1}, {2, 5}, {3, 4}}. + */ + @Test + public void testCaseLargeGraph() { + List edges = Arrays.asList(new int[] {0, 1}, new int[] {1, 2}, new int[] {2, 3}, new int[] {3, 4}, new int[] {4, 5}, new int[] {5, 0}, new int[] {1, 4}, new int[] {2, 5}); + List matching = EdmondsBlossomAlgorithm.maximumMatching(edges, 6); + + // Check if the size of the matching is correct (i.e., 3 pairs) + assertEquals(3, matching.size()); + + // Check that the result contains valid pairs (any order is fine) + // Valid maximum matchings could be {{0, 1}, {2, 5}, {3, 4}} or {{0, 1}, {2, 3}, {4, 5}}, etc. + int[][] possibleMatching1 = new int[][] {{0, 1}, {2, 5}, {3, 4}}; + int[][] possibleMatching2 = new int[][] {{0, 1}, {2, 3}, {4, 5}}; + int[][] result = convertMatchingToArray(matching); + + // Assert that the result is one of the valid maximum matchings + assertTrue(Arrays.deepEquals(result, possibleMatching1) || Arrays.deepEquals(result, possibleMatching2)); + } +}