Extend Graph Algorithms: Added Two Popular Algorithms: BronKerbosch, EdmondsKarp

src/main/java/com/thealgorithms/graph/BronKerbosch.java

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+package com.thealgorithms.graph;
+
+import java.util.ArrayList;
+import java.util.HashSet;
+import java.util.List;
+import java.util.Set;
+
+/**
+ * Implementation of the Bron–Kerbosch algorithm with pivoting for enumerating all maximal cliques
+ * in an undirected graph.
+ *
+ * <p>The input graph is represented as an adjacency list where {@code adjacency.get(u)} returns the
+ * set of vertices adjacent to {@code u}. The algorithm runs in time proportional to the number of
+ * maximal cliques produced and is widely used for clique enumeration problems.</p>
+ *
+ * @author <a href="https://en.wikipedia.org/wiki/Bron%E2%80%93Kerbosch_algorithm">Wikipedia: Bron–Kerbosch algorithm</a>
+ */
+public final class BronKerbosch {
+
+    private BronKerbosch() {
+    }
+
+    /**
+     * Finds all maximal cliques of the provided graph.
+     *
+     * @param adjacency adjacency list where {@code adjacency.size()} equals the number of vertices
+     * @return a list containing every maximal clique, each represented as a {@link Set} of vertices
+     * @throws IllegalArgumentException if the adjacency list is {@code null}, contains {@code null}
+     *         entries, or references invalid vertices
+     */
+    public static List<Set<Integer>> findMaximalCliques(List<Set<Integer>> adjacency) {
+        if (adjacency == null) {
+            throw new IllegalArgumentException("Adjacency list must not be null");
+        }
+
+        int n = adjacency.size();
+        List<Set<Integer>> graph = new ArrayList<>(n);
+        for (int u = 0; u < n; u++) {
+            Set<Integer> neighbors = adjacency.get(u);
+            if (neighbors == null) {
+                throw new IllegalArgumentException("Adjacency list must not contain null sets");
+            }
+            Set<Integer> copy = new HashSet<>();
+            for (int v : neighbors) {
+                if (v < 0 || v >= n) {
+                    throw new IllegalArgumentException("Neighbor index out of bounds: " + v);
+                }
+                if (v != u) {
+                    copy.add(v);
+                }
+            }
+            graph.add(copy);
+        }
+
+        Set<Integer> r = new HashSet<>();
+        Set<Integer> p = new HashSet<>();
+        Set<Integer> x = new HashSet<>();
+        for (int v = 0; v < n; v++) {
+            p.add(v);
+        }
+
+        List<Set<Integer>> cliques = new ArrayList<>();
+        bronKerboschPivot(r, p, x, graph, cliques);
+        return cliques;
+    }
+
+    private static void bronKerboschPivot(Set<Integer> r, Set<Integer> p, Set<Integer> x, List<Set<Integer>> graph, List<Set<Integer>> cliques) {
+        if (p.isEmpty() && x.isEmpty()) {
+            cliques.add(new HashSet<>(r));
+            return;
+        }
+
+        int pivot = choosePivot(p, x, graph);
+        Set<Integer> candidates = new HashSet<>(p);
+        if (pivot != -1) {
+            candidates.removeAll(graph.get(pivot));
+        }
+
+        for (Integer v : candidates) {
+            r.add(v);
+            Set<Integer> newP = intersection(p, graph.get(v));
+            Set<Integer> newX = intersection(x, graph.get(v));
+            bronKerboschPivot(r, newP, newX, graph, cliques);
+            r.remove(v);
+            p.remove(v);
+            x.add(v);
+        }
+    }
+
+    private static int choosePivot(Set<Integer> p, Set<Integer> x, List<Set<Integer>> graph) {
+        int pivot = -1;
+        int maxDegree = -1;
+        Set<Integer> union = new HashSet<>(p);
+        union.addAll(x);
+        for (Integer v : union) {
+            int degree = graph.get(v).size();
+            if (degree > maxDegree) {
+                maxDegree = degree;
+                pivot = v;
+            }
+        }
+        return pivot;
+    }
+
+    private static Set<Integer> intersection(Set<Integer> base, Set<Integer> neighbors) {
+        Set<Integer> result = new HashSet<>();
+        for (Integer v : base) {
+            if (neighbors.contains(v)) {
+                result.add(v);
+            }
+        }
+        return result;
+    }
+}

src/main/java/com/thealgorithms/graph/EdmondsKarp.java

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+package com.thealgorithms.graph;
+
+import java.util.ArrayDeque;
+import java.util.Arrays;
+import java.util.Queue;
+
+/**
+ * Implementation of the Edmonds–Karp algorithm for computing the maximum flow of a directed graph.
+ * <p>
+ * The algorithm runs in O(V * E^2) time and is a specific implementation of the Ford–Fulkerson
+ * method where the augmenting paths are found using breadth-first search (BFS) to ensure the
+ * shortest augmenting paths (in terms of the number of edges) are used.
+ * </p>
+ *
+ * <p>The graph is represented with a capacity matrix where {@code capacity[u][v]} denotes the
+ * capacity of the edge from {@code u} to {@code v}. Negative capacities are not allowed.</p>
+ *
+ * @author <a href="https://en.wikipedia.org/wiki/Edmonds%E2%80%93Karp_algorithm">Wikipedia: EdmondsKarp algorithm</a>
+ */
+public final class EdmondsKarp {
+
+    private EdmondsKarp() {
+    }
+
+    /**
+     * Computes the maximum flow from {@code source} to {@code sink} in the provided capacity matrix.
+     *
+     * @param capacity the capacity matrix representing the directed graph; must be square and non-null
+     * @param source the source vertex index
+     * @param sink the sink vertex index
+     * @return the value of the maximum flow between {@code source} and {@code sink}
+     * @throws IllegalArgumentException if the matrix is {@code null}, not square, contains negative
+     *         capacities, or if {@code source} / {@code sink} indices are invalid
+     */
+    public static int maxFlow(int[][] capacity, int source, int sink) {
+        if (capacity == null || capacity.length == 0) {
+            throw new IllegalArgumentException("Capacity matrix must not be null or empty");
+        }
+
+        final int n = capacity.length;
+        for (int row = 0; row < n; row++) {
+            if (capacity[row] == null || capacity[row].length != n) {
+                throw new IllegalArgumentException("Capacity matrix must be square");
+            }
+            for (int col = 0; col < n; col++) {
+                if (capacity[row][col] < 0) {
+                    throw new IllegalArgumentException("Capacities must be non-negative");
+                }
+            }
+        }
+
+        if (source < 0 || source >= n || sink < 0 || sink >= n) {
+            throw new IllegalArgumentException("Source and sink must be valid vertex indices");
+        }
+        if (source == sink) {
+            return 0;
+        }
+
+        final int[][] residual = new int[n][n];
+        for (int i = 0; i < n; i++) {
+            residual[i] = Arrays.copyOf(capacity[i], n);
+        }
+
+        final int[] parent = new int[n];
+        int maxFlow = 0;
+
+        while (bfs(residual, source, sink, parent)) {
+            int pathFlow = Integer.MAX_VALUE;
+            for (int v = sink; v != source; v = parent[v]) {
+                int u = parent[v];
+                pathFlow = Math.min(pathFlow, residual[u][v]);
+            }
+
+            for (int v = sink; v != source; v = parent[v]) {
+                int u = parent[v];
+                residual[u][v] -= pathFlow;
+                residual[v][u] += pathFlow;
+            }
+
+            maxFlow += pathFlow;
+        }
+
+        return maxFlow;
+    }
+
+    private static boolean bfs(int[][] residual, int source, int sink, int[] parent) {
+        Arrays.fill(parent, -1);
+        parent[source] = source;
+
+        Queue<Integer> queue = new ArrayDeque<>();
+        queue.add(source);
+
+        while (!queue.isEmpty()) {
+            int u = queue.poll();
+            for (int v = 0; v < residual.length; v++) {
+                if (residual[u][v] > 0 && parent[v] == -1) {
+                    parent[v] = u;
+                    if (v == sink) {
+                        return true;
+                    }
+                    queue.add(v);
+                }
+            }
+        }
+        return false;
+    }
+}

src/test/java/com/thealgorithms/graph/BronKerboschTest.java

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+package com.thealgorithms.graph;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import java.util.ArrayList;
+import java.util.HashSet;
+import java.util.List;
+import java.util.Set;
+import org.junit.jupiter.api.DisplayName;
+import org.junit.jupiter.api.Test;
+
+class BronKerboschTest {
+
+    @Test
+    @DisplayName("Complete graph returns single clique")
+    void completeGraph() {
+        List<Set<Integer>> adjacency = buildGraph(4);
+        addUndirectedEdge(adjacency, 0, 1);
+        addUndirectedEdge(adjacency, 0, 2);
+        addUndirectedEdge(adjacency, 0, 3);
+        addUndirectedEdge(adjacency, 1, 2);
+        addUndirectedEdge(adjacency, 1, 3);
+        addUndirectedEdge(adjacency, 2, 3);
+
+        List<Set<Integer>> cliques = BronKerbosch.findMaximalCliques(adjacency);
+        assertEquals(1, cliques.size());
+        assertEquals(Set.of(0, 1, 2, 3), cliques.get(0));
+    }
+
+    @Test
+    @DisplayName("Path graph produces individual edges")
+    void pathGraph() {
+        List<Set<Integer>> adjacency = buildGraph(3);
+        addUndirectedEdge(adjacency, 0, 1);
+        addUndirectedEdge(adjacency, 1, 2);
+
+        List<Set<Integer>> cliques = BronKerbosch.findMaximalCliques(adjacency);
+        Set<Set<Integer>> result = new HashSet<>(cliques);
+        Set<Set<Integer>> expected = Set.of(Set.of(0, 1), Set.of(1, 2));
+        assertEquals(expected, result);
+    }
+
+    @Test
+    @DisplayName("Disconnected graph finds cliques per component")
+    void disconnectedGraph() {
+        List<Set<Integer>> adjacency = buildGraph(5);
+        addUndirectedEdge(adjacency, 0, 1);
+        addUndirectedEdge(adjacency, 0, 2);
+        addUndirectedEdge(adjacency, 1, 2);
+        addUndirectedEdge(adjacency, 3, 4);
+
+        List<Set<Integer>> cliques = BronKerbosch.findMaximalCliques(adjacency);
+        Set<Set<Integer>> result = new HashSet<>(cliques);
+        Set<Set<Integer>> expected = Set.of(Set.of(0, 1, 2), Set.of(3, 4));
+        assertEquals(expected, result);
+    }
+
+    @Test
+    @DisplayName("Null neighbor set triggers exception")
+    void nullNeighborSet() {
+        List<Set<Integer>> adjacency = new ArrayList<>();
+        adjacency.add(null);
+        assertThrows(IllegalArgumentException.class, () -> BronKerbosch.findMaximalCliques(adjacency));
+    }
+
+    private static List<Set<Integer>> buildGraph(int n) {
+        List<Set<Integer>> graph = new ArrayList<>(n);
+        for (int i = 0; i < n; i++) {
+            graph.add(new HashSet<>());
+        }
+        return graph;
+    }
+
+    private static void addUndirectedEdge(List<Set<Integer>> graph, int u, int v) {
+        graph.get(u).add(v);
+        graph.get(v).add(u);
+    }
+}

src/test/java/com/thealgorithms/graph/EdmondsKarpTest.java

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+package com.thealgorithms.graph;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import org.junit.jupiter.api.DisplayName;
+import org.junit.jupiter.api.Test;
+
+class EdmondsKarpTest {
+
+    @Test
+    @DisplayName("Classic CLRS network yields max flow 23")
+    void clrsExample() {
+        int[][] capacity = {{0, 16, 13, 0, 0, 0}, {0, 0, 10, 12, 0, 0}, {0, 4, 0, 0, 14, 0}, {0, 0, 9, 0, 0, 20}, {0, 0, 0, 7, 0, 4}, {0, 0, 0, 0, 0, 0}};
+        int maxFlow = EdmondsKarp.maxFlow(capacity, 0, 5);
+        assertEquals(23, maxFlow);
+    }
+
+    @Test
+    @DisplayName("Disconnected network has zero flow")
+    void disconnectedGraph() {
+        int[][] capacity = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}};
+        int maxFlow = EdmondsKarp.maxFlow(capacity, 0, 2);
+        assertEquals(0, maxFlow);
+    }
+
+    @Test
+    @DisplayName("Source equals sink returns zero")
+    void sourceEqualsSink() {
+        int[][] capacity = {{0, 5}, {0, 0}};
+        int maxFlow = EdmondsKarp.maxFlow(capacity, 0, 0);
+        assertEquals(0, maxFlow);
+    }
+
+    @Test
+    @DisplayName("Invalid matrix throws exception")
+    void invalidMatrix() {
+        int[][] capacity = {{0, 1}, {1}};
+        assertThrows(IllegalArgumentException.class, () -> EdmondsKarp.maxFlow(capacity, 0, 1));
+    }
+
+    @Test
+    @DisplayName("Negative capacity is rejected")
+    void negativeCapacity() {
+        int[][] capacity = {{0, -1}, {0, 0}};
+        assertThrows(IllegalArgumentException.class, () -> EdmondsKarp.maxFlow(capacity, 0, 1));
+    }
+}