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/main/java/com/thealgorithms/graph/GomoryHuTree.java

@@ -0,0 +1,144 @@
+package com.thealgorithms.graph;
+
+import java.util.ArrayDeque;
+import java.util.Arrays;
+import java.util.Queue;
+
+/**
+ * Gomory–Hu tree construction for undirected graphs via n−1 max-flow computations.
+ *
+ * <p>API: {@code buildTree(int[][])} returns {@code {parent, weight}} arrays for the tree.
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Gomory%E2%80%93Hu_tree">Wikipedia: Gomory–Hu tree</a>
+ */
+
+public final class GomoryHuTree {
+    private GomoryHuTree() {
+    }
+
+    public static int[][] buildTree(int[][] cap) {
+        validateCapacityMatrix(cap);
+        final int n = cap.length;
+        if (n == 1) {
+            return new int[][] {new int[] {-1}, new int[] {0}};
+        }
+
+        int[] parent = new int[n];
+        int[] weight = new int[n];
+        Arrays.fill(parent, 0);
+        parent[0] = -1;
+        weight[0] = 0;
+
+        for (int s = 1; s < n; s++) {
+            int t = parent[s];
+            MaxFlowResult res = edmondsKarpWithMinCut(cap, s, t);
+            int f = res.flow;
+            weight[s] = f;
+
+            for (int v = 0; v < n; v++) {
+                if (v != s && parent[v] == t && res.reachable[v]) {
+                    parent[v] = s;
+                }
+            }
+
+            if (t != 0 && res.reachable[parent[t]]) {
+                parent[s] = parent[t];
+                parent[t] = s;
+                weight[s] = weight[t];
+                weight[t] = f;
+            }
+        }
+        return new int[][] {parent, weight};
+    }
+
+    private static void validateCapacityMatrix(int[][] cap) {
+        if (cap == null || cap.length == 0) {
+            throw new IllegalArgumentException("Capacity matrix must not be null or empty");
+        }
+        final int n = cap.length;
+        for (int i = 0; i < n; i++) {
+            if (cap[i] == null || cap[i].length != n) {
+                throw new IllegalArgumentException("Capacity matrix must be square");
+            }
+            for (int j = 0; j < n; j++) {
+                if (cap[i][j] < 0) {
+                    throw new IllegalArgumentException("Capacities must be non-negative");
+                }
+            }
+        }
+    }
+
+    private static final class MaxFlowResult {
+        final int flow;
+        final boolean[] reachable;
+        MaxFlowResult(int flow, boolean[] reachable) {
+            this.flow = flow;
+            this.reachable = reachable;
+        }
+    }
+
+    private static MaxFlowResult edmondsKarpWithMinCut(int[][] capacity, int source, int sink) {
+        final int n = capacity.length;
+        int[][] residual = new int[n][n];
+        for (int i = 0; i < n; i++) {
+            residual[i] = Arrays.copyOf(capacity[i], n);
+        }
+
+        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;
+        }
+
+        boolean[] reachable = new boolean[n];
+        markReachable(residual, source, reachable);
+        return new MaxFlowResult(maxFlow, reachable);
+    }
+
+    private static boolean bfs(int[][] residual, int source, int sink, int[] parent) {
+        Arrays.fill(parent, -1);
+        parent[source] = source;
+        Queue<Integer> q = new ArrayDeque<>();
+        q.add(source);
+        while (!q.isEmpty()) {
+            int u = q.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;
+                    }
+                    q.add(v);
+                }
+            }
+        }
+        return false;
+    }
+
+    private static void markReachable(int[][] residual, int source, boolean[] vis) {
+        Arrays.fill(vis, false);
+        Queue<Integer> q = new ArrayDeque<>();
+        vis[source] = true;
+        q.add(source);
+        while (!q.isEmpty()) {
+            int u = q.poll();
+            for (int v = 0; v < residual.length; v++) {
+                if (!vis[v] && residual[u][v] > 0) {
+                    vis[v] = true;
+                    q.add(v);
+                }
+            }
+        }
+    }
+}

src/main/java/com/thealgorithms/randomized/MonteCarloIntegration.java

@@ -64,13 +64,21 @@ public final class MonteCarloIntegration {
         if (!validate(fx, a, b, n)) {
             throw new IllegalArgumentException("Invalid input parameters");
         }
-        double totalArea = 0.0;
+        double total = 0.0;
         double interval = b - a;
-        for (int i = 0; i < n; i++) {
-            double x = a + generator.nextDouble() * interval;
-            totalArea += fx.apply(x);
+        int pairs = n / 2;
+        for (int i = 0; i < pairs; i++) {
+            double u = generator.nextDouble();
+            double x1 = a + u * interval;
+            double x2 = a + (1.0 - u) * interval;
+            total += fx.apply(x1);
+            total += fx.apply(x2);
         }
-        return interval * totalArea / n;
+        if ((n & 1) == 1) {
+            double x = a + generator.nextDouble() * interval;
+            total += fx.apply(x);
+        }
+        return interval * total / n;
     }
 
     private static boolean validate(Function<Double, Double> fx, double a, double b, int n) {