Create 0085-Maximal-Rectangle

dsa-solutions/lc-solutions/0000-0099/0085-Maximal-Rectangle.md

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+---
+id: Maximal-Rectangle
+title: Maximal Rectangle
+sidebar_label: Maximal Rectangle
+tags:
+  - algorithms
+  - dynamic programming
+  - stack
+  - matrix
+  - binary matrix
+  - rectangle
+---
+
+## Problem Description
+
+| Problem Statement                                       | Solution Link                                                              | LeetCode Profile                                        |
+| :------------------------------------------------------ | :------------------------------------------------------------------------- | :------------------------------------------------------ |
+| [Maximal-Rectangle](https://leetcode.com/problems/Maximal-Rectangle/description/) | [Maximal-Rectangle Solution on LeetCode](https://leetcode.com/problems/Maximal-Rectangle/solutions/) | [Nikita Saini](https://leetcode.com/u/Saini_Nikita/) |
+
+### Problem Description
+
+Given a `rows x cols` binary matrix filled with `0's` and `1's`, find the largest rectangle containing only `1's` and return its area.
+
+### Examples
+
+#### Example 1:
+**Input:** `matrix = [["1","0","1","0","0"],["1","0","1","1","1"],["1","1","1","1","1"],["1","0","0","1","0"]]`
+**Output:** `6`
+**Explanation:** `The maximal rectangle is shown in the above picture.`
+
+#### Example 2:
+**Input:** `matrix = [["0"]]`
+**Output:** `0`
+
+
+#### Example 3:
+**Input:** `matrix = [["1"]]`
+**Output:** `1`
+
+
+### Constraints
+
+- `rows == matrix.length`
+- `cols == matrix[i].length`
+- `1 <= rows, cols <= 200`
+- `matrix[i][j]` is `'0'` or `'1'`.
+
+## Approach
+
+1. **Transform the Matrix into Histograms**: For each row in the matrix, treat each column as the height of a histogram bar.
+2. **Use Histogram Technique**: Apply the largest rectangle in histogram technique for each row.
+3. **Dynamic Update**: Update the heights of the histogram bars dynamically while iterating over rows.
+
+### Step-by-Step Algorithm
+
+1. **Initialization**:
+   - Create a list `heights` of zeros with a length equal to the number of columns in the matrix.
+
+2. **Iterate over Rows**:
+   - For each row, update the `heights` where:
+     - If `matrix[i][j] == '1'`, increment `heights[j]` by 1.
+     - If `matrix[i][j] == '0'`, reset `heights[j]` to 0.
+
+3. **Calculate Maximum Rectangle**:
+   - For each updated `heights` array, use a stack to find the maximum rectangle area in the histogram.
+
+### Python Solution
+
+```python
+def maximalRectangle(matrix):
+    if not matrix:
+        return 0
+    max_area = 0
+    heights = [0] * len(matrix[0])
+
+    for row in matrix:
+        for i in range(len(row)):
+            if row[i] == '1':
+                heights[i] += 1
+            else:
+                heights[i] = 0
+
+        max_area = max(max_area, largestRectangleArea(heights))
+
+    return max_area
+
+def largestRectangleArea(heights):
+    stack = []
+    max_area = 0
+    heights.append(0)
+
+    for i in range(len(heights)):
+        while stack and heights[i] < heights[stack[-1]]:
+            h = heights[stack.pop()]
+            w = i if not stack else i - stack[-1] - 1
+            max_area = max(max_area, h * w)
+        stack.append(i)
+
+    heights.pop()
+    return max_area
+```
+
+### Java Solution
+```java
+import java.util.Stack;
+
+class Solution {
+    public int maximalRectangle(char[][] matrix) {
+        if (matrix.length == 0) return 0;
+        int maxArea = 0;
+        int[] heights = new int[matrix[0].length];
+
+        for (char[] row : matrix) {
+            for (int i = 0; i < row.length; i++) {
+                heights[i] = row[i] == '1' ? heights[i] + 1 : 0;
+            }
+            maxArea = Math.max(maxArea, largestRectangleArea(heights));
+        }
+
+        return maxArea;
+    }
+
+    private int largestRectangleArea(int[] heights) {
+        Stack<Integer> stack = new Stack<>();
+        int maxArea = 0;
+        int[] h = new int[heights.length + 1];
+        System.arraycopy(heights, 0, h, 0, heights.length);
+
+        for (int i = 0; i < h.length; i++) {
+            while (!stack.isEmpty() && h[i] < h[stack.peek()]) {
+                int height = h[stack.pop()];
+                int width = stack.isEmpty() ? i : i - stack.peek() - 1;
+                maxArea = Math.max(maxArea, height * width);
+            }
+            stack.push(i);
+        }
+
+        return maxArea;
+    }
+}
+```
+
+### C++ Solution
+```cpp
+#include <vector>
+#include <stack>
+#include <algorithm>
+
+class Solution {
+public:
+    int maximalRectangle(std::vector<std::vector<char>>& matrix) {
+        if (matrix.empty()) return 0;
+        int maxArea = 0;
+        std::vector<int> heights(matrix[0].size(), 0);
+
+        for (const auto& row : matrix) {
+            for (size_t i = 0; i < row.size(); ++i) {
+                heights[i] = row[i] == '1' ? heights[i] + 1 : 0;
+            }
+            maxArea = std::max(maxArea, largestRectangleArea(heights));
+        }
+
+        return maxArea;
+    }
+
+private:
+    int largestRectangleArea(const std::vector<int>& heights) {
+        std::stack<int> stack;
+        int maxArea = 0;
+        std::vector<int> h = heights;
+        h.push_back(0);
+
+        for (size_t i = 0; i < h.size(); ++i) {
+            while (!stack.empty() && h[i] < h[stack.top()]) {
+                int height = h[stack.top()];
+                stack.pop();
+                int width = stack.empty() ? i : i - stack.top() - 1;
+                maxArea = std::max(maxArea, height * width);
+            }
+            stack.push(i);
+        }
+
+        return maxArea;
+    }
+};
+```
+
+### C Solution
+```c
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+int maximalRectangle(char** matrix, int matrixSize, int* matrixColSize) {
+    if (matrixSize == 0) return 0;
+    int maxArea = 0;
+    int heights[*matrixColSize];
+    memset(heights, 0, sizeof(heights));
+
+    for (int i = 0; i < matrixSize; i++) {
+        for (int j = 0; j < *matrixColSize; j++) {
+            heights[j] = matrix[i][j] == '1' ? heights[j] + 1 : 0;
+        }
+        maxArea = fmax(maxArea, largestRectangleArea(heights, *matrixColSize));
+    }
+
+    return maxArea;
+}
+
+int largestRectangleArea(int* heights, int size) {
+    int maxArea = 0;
+    int* stack = (int*)malloc((size + 1) * sizeof(int));
+    int top = -1;
+    heights[size] = 0;
+
+    for (int i = 0; i <= size; i++) {
+        while (top != -1 && heights[i] < heights[stack[top]]) {
+            int height = heights[stack[top--]];
+            int width = top == -1 ? i : i - stack[top] - 1;
+            maxArea = fmax(maxArea, height * width);
+        }
+        stack[++top] = i;
+    }
+
+    free(stack);
+    return maxArea;
+}
+```
+
+### JavaScript Solution
+```js
+var maximalRectangle = function(matrix) {
+    if (matrix.length === 0) return 0;
+    let maxArea = 0;
+    let heights = new Array(matrix[0].length).fill(0);
+
+    for (let row of matrix) {
+        for (let i = 0; i < row.length; i++) {
+            heights[i] = row[i] === '1' ? heights[i] + 1 : 0;
+        }
+        maxArea = Math.max(maxArea, largestRectangleArea(heights));
+    }
+
+    return maxArea;
+};
+
+var largestRectangleArea = function(heights) {
+    let stack = [];
+    let maxArea = 0;
+    heights.push(0);
+
+    for (let i = 0; i < heights.length; i++) {
+        while (stack.length && heights[i] < heights[stack[stack.length - 1]]) {
+            let height = heights[stack.pop()];
+            let width = stack.length === 0 ? i : i - stack[stack.length - 1] - 1;
+            maxArea = Math.max(maxArea, height * width);
+        }
+        stack.push(i);
+    }
+
+    heights.pop();
+    return maxArea;
+};
+```
+
+### Conclusion
+This problem combines concepts from dynamic programming and stack-based approaches to efficiently calculate the largest rectangle containing only 1's in a binary matrix. By converting the problem to a series of histogram problems and using an optimized approach to find the largest rectangle in a histogram, we achieve a time-efficient solution.
+
+