Feat/segment tree 2d (#7363)

* feat: add 2D segment tree implementation

* test: add comprehensive unit tests for 2D segment tree

* style: format code using clang-format

Author: YassaTalaat80

src/main/java/com/thealgorithms/datastructures/trees/SegmentTree2D.java

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+package com.thealgorithms.datastructures.trees;
+
+/**
+ * 2D Segment Tree (Tree of Trees) implementation.
+ * This data structure supports point updates and submatrix sum queries
+ * in a 2D grid. It achieves this by nesting 1D Segment Trees within a 1D Segment Tree.
+ *
+ * Time Complexity:
+ * - Build/Initialization: O(N * M)
+ * - Point Update: O(log N * log M)
+ * - Submatrix Query: O(log N * log M)
+ *
+ * @see <a href="https://cp-algorithms.com/data_structures/segment_tree.html#2d-segment-tree">2D Segment Tree</a>
+ */
+public class SegmentTree2D {
+
+    /**
+     * Represents a 1D Segment Tree.
+     * This is equivalent to your 'Sagara' struct. It manages the columns (X-axis).
+     */
+    public static class SegmentTree1D {
+        private int n;
+        private final int[] tree;
+
+        /**
+         * Initializes the 1D Segment Tree with the nearest power of 2.
+         *
+         * @param size The expected number of elements (columns).
+         */
+        public SegmentTree1D(int size) {
+            n = 1;
+            while (n < size) {
+                n *= 2;
+            }
+            tree = new int[n * 2];
+        }
+
+        /**
+         * Recursively updates a point in the 1D tree.
+         */
+        private void update(int index, int val, int node, int lx, int rx) {
+            if (rx - lx == 1) {
+                tree[node] = val;
+                return;
+            }
+
+            int mid = lx + (rx - lx) / 2;
+            int leftChild = node * 2 + 1;
+            int rightChild = node * 2 + 2;
+
+            if (index < mid) {
+                update(index, val, leftChild, lx, mid);
+            } else {
+                update(index, val, rightChild, mid, rx);
+            }
+
+            tree[node] = tree[leftChild] + tree[rightChild];
+        }
+
+        /**
+         * Public wrapper to update a specific index.
+         *
+         * @param index The column index to update.
+         * @param val   The new value.
+         */
+        public void update(int index, int val) {
+            update(index, val, 0, 0, n);
+        }
+
+        /**
+         * Retrieves the exact value at a specific leaf node.
+         *
+         * @param index The column index.
+         * @return The value at the given index.
+         */
+        public int get(int index) {
+            return query(index, index + 1, 0, 0, n);
+        }
+
+        /**
+         * Recursively queries the sum in a 1D range.
+         */
+        private int query(int l, int r, int node, int lx, int rx) {
+            if (lx >= r || rx <= l) {
+                return 0; // Out of bounds
+            }
+            if (lx >= l && rx <= r) {
+                return tree[node]; // Fully inside
+            }
+
+            int mid = lx + (rx - lx) / 2;
+            int leftSum = query(l, r, node * 2 + 1, lx, mid);
+            int rightSum = query(l, r, node * 2 + 2, mid, rx);
+
+            return leftSum + rightSum;
+        }
+
+        /**
+         * Public wrapper to query the sum in the range [l, r).
+         *
+         * @param l Left boundary (inclusive).
+         * @param r Right boundary (exclusive).
+         * @return The sum of the range.
+         */
+        public int query(int l, int r) {
+            return query(l, r, 0, 0, n);
+        }
+    }
+
+    // --- Start of 2D Segment Tree (equivalent to 'Sagara2D') ---
+
+    private int n;
+    private final SegmentTree1D[] tree;
+
+    /**
+     * Initializes the 2D Segment Tree.
+     *
+     * @param rows The number of rows in the matrix.
+     * @param cols The number of columns in the matrix.
+     */
+    public SegmentTree2D(int rows, int cols) {
+        n = 1;
+        while (n < rows) {
+            n *= 2;
+        }
+        tree = new SegmentTree1D[n * 2];
+        for (int i = 0; i < n * 2; i++) {
+            // Every node in the outer tree is a full 1D tree!
+            tree[i] = new SegmentTree1D(cols);
+        }
+    }
+
+    /**
+     * Recursively updates a point in the 2D grid.
+     */
+    private void update(int row, int col, int val, int node, int lx, int rx) {
+        if (rx - lx == 1) {
+            tree[node].update(col, val);
+            return;
+        }
+
+        int mid = lx + (rx - lx) / 2;
+        int leftChild = node * 2 + 1;
+        int rightChild = node * 2 + 2;
+
+        if (row < mid) {
+            update(row, col, val, leftChild, lx, mid);
+        } else {
+            update(row, col, val, rightChild, mid, rx);
+        }
+
+        // The value of the current node's column is the sum of its children's column values
+        int leftVal = tree[leftChild].get(col);
+        int rightVal = tree[rightChild].get(col);
+        tree[node].update(col, leftVal + rightVal);
+    }
+
+    /**
+     * Public wrapper to update a specific point (row, col).
+     *
+     * @param row The row index.
+     * @param col The column index.
+     * @param val The new value.
+     */
+    public void update(int row, int col, int val) {
+        update(row, col, val, 0, 0, n);
+    }
+
+    /**
+     * Recursively queries the sum in a submatrix.
+     */
+    private int query(int top, int bottom, int left, int right, int node, int lx, int rx) {
+        if (lx >= bottom || rx <= top) {
+            return 0; // Out of bounds
+        }
+        if (lx >= top && rx <= bottom) {
+            // Fully inside the row range, so delegate the column query to the 1D tree
+            return tree[node].query(left, right);
+        }
+
+        int mid = lx + (rx - lx) / 2;
+        int leftSum = query(top, bottom, left, right, node * 2 + 1, lx, mid);
+        int rightSum = query(top, bottom, left, right, node * 2 + 2, mid, rx);
+
+        return leftSum + rightSum;
+    }
+
+    /**
+     * Public wrapper to query the sum of a submatrix.
+     * Note: boundaries are [top, bottom) and [left, right).
+     *
+     * @param top    Top row index (inclusive).
+     * @param bottom Bottom row index (exclusive).
+     * @param left   Left column index (inclusive).
+     * @param right  Right column index (exclusive).
+     * @return The sum of the submatrix.
+     */
+    public int query(int top, int bottom, int left, int right) {
+        return query(top, bottom, left, right, 0, 0, n);
+    }
+}

src/test/java/com/thealgorithms/datastructures/trees/SegmentTree2DTest.java

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+package com.thealgorithms.datastructures.trees;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+
+import org.junit.jupiter.api.Test;
+
+public class SegmentTree2DTest {
+
+    @Test
+    void testInitialEmptyQueries() {
+        SegmentTree2D segmentTree = new SegmentTree2D(4, 4);
+
+        // Initial tree should return 0 for any query
+        assertEquals(0, segmentTree.query(0, 4, 0, 4));
+        assertEquals(0, segmentTree.query(1, 3, 1, 3));
+    }
+
+    @Test
+    void testUpdateAndPointQuery() {
+        SegmentTree2D segmentTree = new SegmentTree2D(5, 5);
+
+        segmentTree.update(2, 3, 10);
+        segmentTree.update(0, 0, 5);
+
+        // Querying single points [row, row+1) x [col, col+1)
+        assertEquals(10, segmentTree.query(2, 3, 3, 4));
+        assertEquals(5, segmentTree.query(0, 1, 0, 1));
+
+        // Empty point should be 0
+        assertEquals(0, segmentTree.query(1, 2, 1, 2));
+    }
+
+    @Test
+    void testSubmatrixQuery() {
+        SegmentTree2D segmentTree = new SegmentTree2D(4, 4);
+
+        // Matrix simulation:
+        // [1, 2, 0, 0]
+        // [3, 4, 0, 0]
+        // [0, 0, 0, 0]
+        // [0, 0, 0, 0]
+        segmentTree.update(0, 0, 1);
+        segmentTree.update(0, 1, 2);
+        segmentTree.update(1, 0, 3);
+        segmentTree.update(1, 1, 4);
+
+        // Top-left 2x2 sum: 1+2+3+4 = 10
+        assertEquals(10, segmentTree.query(0, 2, 0, 2));
+
+        // First row sum: 1+2 = 3
+        assertEquals(3, segmentTree.query(0, 1, 0, 4));
+
+        // Second column sum: 2+4 = 6
+        assertEquals(6, segmentTree.query(0, 4, 1, 2));
+    }
+
+    @Test
+    void testUpdateOverwriting() {
+        SegmentTree2D segmentTree = new SegmentTree2D(3, 3);
+
+        segmentTree.update(1, 1, 5);
+        assertEquals(5, segmentTree.query(1, 2, 1, 2));
+
+        // Overwrite the same point
+        segmentTree.update(1, 1, 20);
+        assertEquals(20, segmentTree.query(1, 2, 1, 2));
+
+        // Full matrix sum should just be this point
+        assertEquals(20, segmentTree.query(0, 3, 0, 3));
+    }
+}