feat: Added Gomory–Hu Tree (all-pairs min-cuts via n1 max-flows) (#6818)

* Add Gomory–Hu Tree (all-pairs min-cuts via n1 max-flows)

* Stabilize Monte Carlo integration with antithetic variates to reduce variance

* Fix Checkstyle in GomoryHuTreeTest: remove inner assignments, add braces, split declarations

* SpotBugs: use RandomGenerator interface in test helper

---------

Co-authored-by: Deniz Altunkapan <deniz.altunkapan@outlook.com>

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

@@ -0,0 +1,132 @@
+package com.thealgorithms.graph;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+
+import java.util.ArrayDeque;
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.List;
+import java.util.Queue;
+import java.util.Random;
+import java.util.random.RandomGenerator;
+import org.junit.jupiter.api.DisplayName;
+import org.junit.jupiter.api.Test;
+
+class GomoryHuTreeTest {
+
+    @Test
+    @DisplayName("Single node graph")
+    void singleNode() {
+        int[][] cap = {{0}};
+        int[][] res = GomoryHuTree.buildTree(cap);
+        int[] parent = res[0];
+        int[] weight = res[1];
+        assertEquals(-1, parent[0]);
+        assertEquals(0, weight[0]);
+    }
+
+    @Test
+    @DisplayName("Triangle undirected graph with known min-cuts")
+    void triangleGraph() {
+        // 0-1:3, 1-2:2, 0-2:4
+        int[][] cap = new int[3][3];
+        cap[0][1] = 3;
+        cap[1][0] = 3;
+        cap[1][2] = 2;
+        cap[2][1] = 2;
+        cap[0][2] = 4;
+        cap[2][0] = 4;
+
+        int[][] tree = GomoryHuTree.buildTree(cap);
+        // validate all pairs via path-min-edge equals maxflow
+        validateAllPairs(cap, tree);
+    }
+
+    @Test
+    @DisplayName("Random small undirected graphs compare to EdmondsKarp")
+    void randomSmallGraphs() {
+        Random rng = new Random(42);
+        for (int n = 2; n <= 6; n++) {
+            for (int iter = 0; iter < 10; iter++) {
+                int[][] cap = randSymmetricMatrix(n, 0, 5, rng);
+                int[][] tree = GomoryHuTree.buildTree(cap);
+                validateAllPairs(cap, tree);
+            }
+        }
+    }
+
+    private static int[][] randSymmetricMatrix(int n, int lo, int hi, RandomGenerator rng) {
+        int[][] a = new int[n][n];
+        for (int i = 0; i < n; i++) {
+            for (int j = i + 1; j < n; j++) {
+                int w = rng.nextInt(hi - lo + 1) + lo;
+                a[i][j] = w;
+                a[j][i] = w;
+            }
+        }
+        // zero diagonal
+        for (int i = 0; i < n; i++) {
+            a[i][i] = 0;
+        }
+        return a;
+    }
+
+    private static void validateAllPairs(int[][] cap, int[][] tree) {
+        int n = cap.length;
+        int[] parent = tree[0];
+        int[] weight = tree[1];
+
+        // build adjacency list of tree without generic array creation
+        List<List<int[]>> g = new ArrayList<>();
+        for (int i = 0; i < n; i++) {
+            g.add(new ArrayList<>());
+        }
+        for (int v = 1; v < n; v++) {
+            int u = parent[v];
+            int w = weight[v];
+            g.get(u).add(new int[] {v, w});
+            g.get(v).add(new int[] {u, w});
+        }
+
+        for (int s = 0; s < n; s++) {
+            for (int t = s + 1; t < n; t++) {
+                int treeVal = minEdgeOnPath(g, s, t);
+                int flowVal = EdmondsKarp.maxFlow(cap, s, t);
+                assertEquals(flowVal, treeVal, "pair (" + s + "," + t + ")");
+            }
+        }
+    }
+
+    private static int minEdgeOnPath(List<List<int[]>> g, int s, int t) {
+        // BFS to record parent and edge weight along the path, since it's a tree, unique path exists
+        int n = g.size();
+        int[] parent = new int[n];
+        int[] edgeW = new int[n];
+        Arrays.fill(parent, -1);
+        Queue<Integer> q = new ArrayDeque<>();
+        q.add(s);
+        parent[s] = s;
+        while (!q.isEmpty()) {
+            int u = q.poll();
+            if (u == t) {
+                break;
+            }
+            for (int[] e : g.get(u)) {
+                int v = e[0];
+                int w = e[1];
+                if (parent[v] == -1) {
+                    parent[v] = u;
+                    edgeW[v] = w;
+                    q.add(v);
+                }
+            }
+        }
+        int cur = t;
+        int ans = Integer.MAX_VALUE;
+        while (cur != s) {
+            ans = Math.min(ans, edgeW[cur]);
+            cur = parent[cur];
+        }
+        return ans == Integer.MAX_VALUE ? 0 : ans;
+    }
+}