TheAlgorithms · DenizAltunkapan · Aug 14, 2025 · Aug 13, 2025 · Aug 13, 2025 · Aug 13, 2025
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+package com.thealgorithms.graph;
+
+import java.util.ArrayDeque;
+import java.util.Arrays;
+import java.util.List;
+import java.util.Queue;
+
+/**
+ * Hopcroft–Karp algorithm for maximum bipartite matching.
+ *
+ * Left part: vertices [0,nLeft-1], Right part: [0,nRight-1].
+ * Adjacency list: for each left vertex u, list right vertices v it connects to.
+ *
+ * Time complexity: O(E * sqrt(V)).
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm">
+ *      Wikipedia: Hopcroft–Karp algorithm</a>
+ * @author ptzecher
+ */
+public class HopcroftKarp {
+
+    private final int nLeft;
+    private final List<List<Integer>> adj;
+
+    private final int[] pairU;
+    private final int[] pairV;
+    private final int[] dist;
+
+    public HopcroftKarp(int nLeft, int nRight, List<List<Integer>> adj) {
+        this.nLeft = nLeft;
+        this.adj = adj;
+
+        this.pairU = new int[nLeft];
+        this.pairV = new int[nRight];
+        this.dist = new int[nLeft];
+
+        Arrays.fill(pairU, -1);
+        Arrays.fill(pairV, -1);
+    }
+
+    /** Returns the size of the maximum matching. */
+    public int maxMatching() {
+        int matching = 0;
+        while (bfs()) {
+            for (int u = 0; u < nLeft; u++) {
+                if (pairU[u] == -1 && dfs(u)) {
+                    matching++;
+                }
+            }
+        }
+        return matching;
+    }
+
+    // BFS to build layers
+    private boolean bfs() {
+        Queue<Integer> queue = new ArrayDeque<>();
+        Arrays.fill(dist, -1);
+
+        for (int u = 0; u < nLeft; u++) {
+            if (pairU[u] == -1) {
+                dist[u] = 0;
+                queue.add(u);
+            }
+        }
+
+        boolean foundAugPath = false;
+        while (!queue.isEmpty()) {
+            int u = queue.poll();
+            for (int v : adj.get(u)) {
+                int matchedLeft = pairV[v];
+                if (matchedLeft == -1) {
+                    foundAugPath = true;
+                } else if (dist[matchedLeft] == -1) {
+                    dist[matchedLeft] = dist[u] + 1;
+                    queue.add(matchedLeft);
+                }
+            }
+        }
+        return foundAugPath;
+    }
+
+    // DFS to find augmenting paths within the BFS layering
+    private boolean dfs(int u) {
+        for (int v : adj.get(u)) {
+            int matchedLeft = pairV[v];
+            if (matchedLeft == -1 || (dist[matchedLeft] == dist[u] + 1 && dfs(matchedLeft))) {
+                pairU[u] = v;
+                pairV[v] = u;
+                return true;
+            }
+        }
+        dist[u] = -1;
+        return false;
+    }
+
+    public int[] getLeftMatches() {
+        return pairU.clone();
+    }
+
+    public int[] getRightMatches() {
+        return pairV.clone();
+    }
+}
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+package com.thealgorithms.graph;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertFalse;
+
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.List;
+import org.junit.jupiter.api.DisplayName;
+import org.junit.jupiter.api.Test;
+
+/**
+ * Unit tests for Hopcroft–Karp algorithm.
+ *
+ * @author ptzecher
+ */
+class HopcroftKarpTest {
+
+    private static List<List<Integer>> adj(int nLeft) {
+        List<List<Integer>> g = new ArrayList<>(nLeft);
+        for (int i = 0; i < nLeft; i++) { // braces required by Checkstyle
+            g.add(new ArrayList<>());
+        }
+        return g;
+    }
+
+    @Test
+    @DisplayName("Empty graph has matching 0")
+    void emptyGraph() {
+        List<List<Integer>> g = adj(3);
+        HopcroftKarp hk = new HopcroftKarp(3, 4, g);
+        assertEquals(0, hk.maxMatching());
+    }
+
+    @Test
+    @DisplayName("Single edge gives matching 1")
+    void singleEdge() {
+        List<List<Integer>> g = adj(1);
+        g.get(0).add(0);
+        HopcroftKarp hk = new HopcroftKarp(1, 1, g);
+        assertEquals(1, hk.maxMatching());
+
+        int[] leftMatch = hk.getLeftMatches();
+        int[] rightMatch = hk.getRightMatches();
+        assertEquals(0, leftMatch[0]);
+        assertEquals(0, rightMatch[0]);
+    }
+
+    @Test
+    @DisplayName("Disjoint edges match perfectly")
+    void disjointEdges() {
+        // L0-R0, L1-R1, L2-R2
+        List<List<Integer>> g = adj(3);
+        g.get(0).add(0);
+        g.get(1).add(1);
+        g.get(2).add(2);
+
+        HopcroftKarp hk = new HopcroftKarp(3, 3, g);
+        assertEquals(3, hk.maxMatching());
+
+        int[] leftMatch = hk.getLeftMatches();
+        int[] rightMatch = hk.getRightMatches();
+        for (int i = 0; i < 3; i++) {
+            assertEquals(i, leftMatch[i]);
+            assertEquals(i, rightMatch[i]);
+        }
+    }
+
+    @Test
+    @DisplayName("Complete bipartite K(3,4) matches min(3,4)=3")
+    void completeK34() {
+        int nLeft = 3;
+        int nRight = 4; // split declarations
+        List<List<Integer>> g = adj(nLeft);
+        for (int u = 0; u < nLeft; u++) {
+            g.get(u).addAll(Arrays.asList(0, 1, 2, 3));
+        }
+        HopcroftKarp hk = new HopcroftKarp(nLeft, nRight, g);
+        assertEquals(3, hk.maxMatching());
+
+        // sanity: no two left vertices share the same matched right vertex
+        int[] leftMatch = hk.getLeftMatches();
+        boolean[] used = new boolean[nRight];
+        for (int u = 0; u < nLeft; u++) {
+            int v = leftMatch[u];
+            if (v != -1) {
+                assertFalse(used[v]);
+                used[v] = true;
+            }
+        }
+    }
+
+    @Test
+    @DisplayName("Rectangular, sparse graph")
+    void rectangularSparse() {
+        // Left: 5, Right: 2
+        // edges: L0-R0, L1-R1, L2-R0, L3-R1  (max matching = 2)
+        List<List<Integer>> g = adj(5);
+        g.get(0).add(0);
+        g.get(1).add(1);
+        g.get(2).add(0);
+        g.get(3).add(1);
+        // L4 isolated
+
+        HopcroftKarp hk = new HopcroftKarp(5, 2, g);
+        assertEquals(2, hk.maxMatching());
+
+        int[] leftMatch = hk.getLeftMatches();
+        int[] rightMatch = hk.getRightMatches();
+
+        // Check consistency: if leftMatch[u]=v then rightMatch[v]=u
+        for (int u = 0; u < 5; u++) {
+            int v = leftMatch[u];
+            if (v != -1) {
+                assertEquals(u, rightMatch[v]);
+            }
+        }
+    }
+
+    @Test
+    @DisplayName("Layering advantage case (short augmenting paths)")
+    void layeringAdvantage() {
+        // Left 4, Right 4
+        List<List<Integer>> g = adj(4);
+        g.get(0).addAll(Arrays.asList(0, 1));
+        g.get(1).addAll(Arrays.asList(1, 2));
+        g.get(2).addAll(Arrays.asList(2, 3));
+        g.get(3).addAll(Arrays.asList(0, 3));
+
+        HopcroftKarp hk = new HopcroftKarp(4, 4, g);
+        assertEquals(4, hk.maxMatching()); // perfect matching exists
+    }
+}