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algorithms/000407-trapping-rain-water-ii/README.md

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+407\. Trapping Rain Water II
+
+**Difficulty:** Hard
+
+**Topics:** `Array`, `Breadth-First Search`, `Heap (Priority Queue)`, `Matrix`
+
+Given an `m x n` integer matrix `heightMap` representing the height of each unit cell in a 2D elevation map, return _the volume of water it can trap after raining_.
+
+**Example 1:**
+
+![trap1-3d](https://assets.leetcode.com/uploads/2021/04/08/trap1-3d.jpg)
+
+- **Input:** heightMap = [[1,4,3,1,3,2],[3,2,1,3,2,4],[2,3,3,2,3,1]]
+- **Output:** 4
+- **Explanation:** After the rain, water is trapped between the blocks.
+  We have two small ponds 1 and 3 units trapped.
+  The total volume of water trapped is 4.
+
+**Example 2:**
+
+![trap2-3d](https://assets.leetcode.com/uploads/2021/04/08/trap2-3d.jpg)
+
+- **Input:** heightMap = [[3,3,3,3,3],[3,2,2,2,3],[3,2,1,2,3],[3,2,2,2,3],[3,3,3,3,3]]
+- **Output:** 10
+
+
+
+**Constraints:**
+
+- `m == heightMap.length`
+- `n == heightMap[i].length`
+- `1 <= m, n <= 200`
+- <code>0 <= heightMap[i][j] <= 2 * 10<sup>4</sup></code>
+
+
+**Solution:**
+
+The "Trapping Rain Water II" problem is a challenging computational problem that requires us to compute the volume of water trapped after raining on a 2D elevation map represented as a matrix. This problem extends the classic "Trapping Rain Water" problem to two dimensions, making the solution more complex due to the need to consider water flow in all directions.
+
+
+## **Key Points**
+1. **Matrix Representation**: The `heightMap` matrix contains the elevation of each cell.
+2. **Boundary Constraints**: Water cannot flow out of the boundary cells.
+3. **Heap Data Structure**: A min-heap (priority queue) is used to simulate the water levels dynamically.
+4. **Visited Matrix**: To prevent revisiting cells, we track visited nodes.
+
+
+## **Approach**
+The solution leverages a **Breadth-First Search (BFS)** approach, guided by a **priority queue (min-heap)**:
+1. Add all boundary cells to the min-heap and mark them as visited.
+2. Process cells in increasing height order:
+   - For each cell, attempt to "trap" water in its neighbors.
+   - Push neighboring cells into the heap with updated height values.
+3. Accumulate the trapped water volume based on the height difference between the current cell and its neighbors.
+
+
+## **Plan**
+1. **Initialization**:
+   - Define matrix dimensions and edge cases.
+   - Initialize a min-heap for boundary cells.
+   - Create a `visited` matrix.
+
+2. **Insert Boundary Cells**:
+   - Push all boundary cells into the heap with their height values.
+   - Mark them as visited.
+
+3. **BFS Traversal**:
+   - While the heap is not empty, extract the cell with the smallest height.
+   - Check all its neighbors and calculate trapped water:
+      - If the neighbor is lower, the height difference contributes to trapped water.
+      - Update the neighbor's height to the current cell's height if higher.
+   - Push the neighbor into the heap and mark it visited.
+
+4. **Return Result**:
+   - The accumulated water volume represents the trapped rainwater.
+
+Let's implement this solution in PHP: **[407. Trapping Rain Water II](https://github.com/mah-shamim/leet-code-in-php/tree/main/algorithms/000407-trapping-rain-water-ii/solution.php)**
+
+```php
+<?php
+/**
+ * @param Integer[][] $heightMap
+ * @return Integer
+ */
+function trapRainWater($heightMap) {
+    ...
+    ...
+    ...
+    /**
+     * go to ./solution.php
+     */
+}
+
+// Example Usage
+$heightMap1 = [[1, 4, 3, 1, 3, 2], [3, 2, 1, 3, 2, 4], [2, 3, 3, 2, 3, 1]];
+$heightMap2 = [[3, 3, 3, 3, 3], [3, 2, 2, 2, 3], [3, 2, 1, 2, 3], [3, 2, 2, 2, 3], [3, 3, 3, 3, 3]];
+
+echo trapRainWater($heightMap1) . "\n"; // Output: 4
+echo trapRainWater($heightMap2) . "\n"; // Output: 10
+?>
+```
+
+### Explanation:
+
+1. **Boundary Initialization**:
+   - All boundary cells are added to the heap to form the outer walls of the container.
+
+2. **Heap Extraction**:
+   - Extract the cell with the lowest height, ensuring that water flows only outward, not inward.
+
+3. **Neighbor Exploration**:
+   - For each neighbor:
+      - Check if it’s within bounds and unvisited.
+      - Calculate the water trapped as `max(0, currentHeight - neighborHeight)`.
+      - Push the updated neighbor height into the heap.
+
+4. **Accumulate Water**:
+   - Add the trapped water for each neighbor to the total.
+
+
+## **Example Walkthrough**
+### Input:
+```php
+$heightMap = [
+    [1, 4, 3, 1, 3, 2],
+    [3, 2, 1, 3, 2, 4],
+    [2, 3, 3, 2, 3, 1]
+];
+```
+### Steps:
+1. **Boundary Cells**:
+   - Push cells from the boundary into the heap with their heights:
+      - Example: `(0, 0, 1)`, `(0, 1, 4)`, etc.
+
+2. **Heap Traversal**:
+   - Extract cell `(0, 0, 1)` (lowest height).
+   - Check neighbors, calculate water trapped:
+      - For neighbor `(1, 0)`: Water trapped = `max(0, 1 - 3) = 0`.
+
+3. **Trapped Water**:
+   - Continue processing until all cells are visited:
+      - Total trapped water = `4`.
+
+
+## **Time Complexity**
+1. **Heap Operations**:
+   - Each cell is pushed and popped from the heap once: `O(m * n * log(m * n))`.
+
+2. **Neighbor Iteration**:
+   - Each cell has up to 4 neighbors: `O(m * n)`.
+
+### **Total Complexity**:
+**`O(m * n * log(m * n))`**
+
+
+## **Output for Example**
+```php
+$heightMap = [
+    [1, 4, 3, 1, 3, 2],
+    [3, 2, 1, 3, 2, 4],
+    [2, 3, 3, 2, 3, 1]
+];
+echo trapRainWater($heightMap); // Output: 4
+```
+
+The "Trapping Rain Water II" problem demonstrates the power of advanced data structures like priority queues combined with BFS. By simulating the flow of water in a 2D elevation map, we can efficiently compute the total trapped water. This solution is optimal for handling large matrices due to its logarithmic heap operations.
+
+**Contact Links**
+
+If you found this series helpful, please consider giving the **[repository](https://github.com/mah-shamim/leet-code-in-php)** a star on GitHub or sharing the post on your favorite social networks 😍. Your support would mean a lot to me!
+
+If you want more helpful content like this, feel free to follow me:
+
+- **[LinkedIn](https://www.linkedin.com/in/arifulhaque/)**
+- **[GitHub](https://github.com/mah-shamim)**

algorithms/000407-trapping-rain-water-ii/solution.php

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+<?php
+
+class Solution {
+
+    /**
+     * @param Integer[][] $heightMap
+     * @return Integer
+     */
+    function trapRainWater($heightMap) {
+        $m = count($heightMap);
+        $n = count($heightMap[0]);
+
+        if ($m < 3 || $n < 3) {
+            return 0;
+        }
+
+        $directions = [[1, 0], [-1, 0], [0, 1], [0, -1]];
+        $minHeap = new SplPriorityQueue();
+        $visited = array_fill(0, $m, array_fill(0, $n, false));
+
+        // Add the boundary cells to the priority queue
+        for ($i = 0; $i < $m; $i++) {
+            for ($j = 0; $j < $n; $j++) {
+                if ($i == 0 || $i == $m - 1 || $j == 0 || $j == $n - 1) {
+                    $minHeap->insert([$i, $j, $heightMap[$i][$j]], -$heightMap[$i][$j]);
+                    $visited[$i][$j] = true;
+                }
+            }
+        }
+
+        $waterTrapped = 0;
+
+        while (!$minHeap->isEmpty()) {
+            list($x, $y, $h) = $minHeap->extract();
+            foreach ($directions as $direction) {
+                $nx = $x + $direction[0];
+                $ny = $y + $direction[1];
+
+                if ($nx >= 0 && $nx < $m && $ny >= 0 && $ny < $n && !$visited[$nx][$ny]) {
+                    $visited[$nx][$ny] = true;
+                    $waterTrapped += max(0, $h - $heightMap[$nx][$ny]);
+                    $minHeap->insert([$nx, $ny, max($h, $heightMap[$nx][$ny])], -max($h, $heightMap[$nx][$ny]));
+                }
+            }
+        }
+
+        return $waterTrapped;
+    }
+}
+/**
+ * @param Integer[][] $heightMap
+ * @return Integer
+ */
+function trapRainWater($heightMap) {
+    $m = count($heightMap);
+    $n = count($heightMap[0]);
+
+    if ($m < 3 || $n < 3) {
+        return 0;
+    }
+
+    $directions = [[1, 0], [-1, 0], [0, 1], [0, -1]];
+    $minHeap = new SplPriorityQueue();
+    $visited = array_fill(0, $m, array_fill(0, $n, false));
+
+    // Add the boundary cells to the priority queue
+    for ($i = 0; $i < $m; $i++) {
+        for ($j = 0; $j < $n; $j++) {
+            if ($i == 0 || $i == $m - 1 || $j == 0 || $j == $n - 1) {
+                $minHeap->insert([$i, $j, $heightMap[$i][$j]], -$heightMap[$i][$j]);
+                $visited[$i][$j] = true;
+            }
+        }
+    }
+
+    $waterTrapped = 0;
+
+    while (!$minHeap->isEmpty()) {
+        list($x, $y, $h) = $minHeap->extract();
+        foreach ($directions as $direction) {
+            $nx = $x + $direction[0];
+            $ny = $y + $direction[1];
+
+            if ($nx >= 0 && $nx < $m && $ny >= 0 && $ny < $n && !$visited[$nx][$ny]) {
+                $visited[$nx][$ny] = true;
+                $waterTrapped += max(0, $h - $heightMap[$nx][$ny]);
+                $minHeap->insert([$nx, $ny, max($h, $heightMap[$nx][$ny])], -max($h, $heightMap[$nx][$ny]));
+            }
+        }
+    }
+
+    return $waterTrapped;
+}