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# Maximal Square

`Dynamic Programming`

## Question

Given a 2D binary matrix filled with 0's and 1's, find the largest square containing all 1's and return its area.

**Example**

For example, given the following matrix:

```
1 0 1 0 0
1 0 1 1 1
1 1 1 1 1
1 0 0 1 0
```

Return 4.

**Tags**

Dynamic Programming Airbnb Facebook

**Related Problems**

Hard Maximal Rectangle

## Analysis

考虑以`f[i][j]`为右下角顶点可以拓展的正方形边长，那么可以由`f[i-1][j]`，`f[i][j-1]`，`f[i-1][j-1]`三者的最小值决定`f[i][j]`的最长边长。

> 当我们判断以某个点为正方形右下角时最大的正方形时，那它的上方，左方和左上方三个点也一定是某个正方形的右下角，否则该点为右下角的正方形最大就是它自己了。这是定性的判断，那具体的最大正方形边长呢？我们知道，该点为右下角的正方形的最大边长，最多比它的上方，左方和左上方为右下角的正方形的边长多1，最好的情况是是它的上方，左方和左上方为右下角的正方形的大小都一样的，这样加上该点就可以构成一个更大的正方形。但如果它的上方，左方和左上方为右下角的正方形的大小不一样，合起来就会缺了某个角落，这时候只能取那三个正方形中最小的正方形的边长加1了。假设`dp[i][j]`表示以i,j为右下角的正方形的最大边长，则有
>
> ```
> dp[i][j] = min(dp[i-1][j-1], dp[i-1][j], dp[i][j-1]) + 1
> ```
>
> 当然，如果这个点在原矩阵中本身就是0的话，那`dp[i][j]`肯定就是0了。

时间`O(mn)`， 空间`O(mn)`

**DP四要素**

* 状态 State
  * `f[i][j]` 表示以i和j作为正方形右下角可以拓展的最大边长
* 方程 Function
  * if `matrix[i][j] == 1`
    * `f[i][j] = min(f[i - 1][j], f[i][j-1], f[i-1][j-1]) + 1;`
  * if `matrix[i][j] == 0`
    * `f[i][j] = 0`
* 初始化 Intialization
  * `f[i][0] = matrix[i][0];`
  * `f[0][j] = matrix[0][j];`
* 答案 Answer
  * `max{f[i][j]}`

**可以使用二维滚动数组优化:**

1. 状态 State
   * `f[i][j]` 表示以i和j作为正方形右下角可以拓展的最大边长
2. 方程 Function
   * `if matrix[i][j] == 1`
     * `f[i%2][j] = min(f[(i - 1)%2][j], f[i%2][j-1], f[(i-1)%2][j-1]) + 1;`
   * `if matrix[i][j] == 0`
   * `f[i%2][j] = 0`
3. 初始化 Intialization
   * `f[i%2][0] = matrix[i][0];`
   * `f[0][j] = matrix[0][j];`
4. 答案 Answer
   * `max{f[i%2][j]}`

## Solution

### Standard Dynamic Programming

```java
public class Solution {
    /**
     * @param matrix: a matrix of 0 and 1
     * @return: an integer
     */
    public int maxSquare(int[][] matrix) {
        if (matrix.length == 0) return 0;

        int m = matrix.length, n = matrix[0].length;
        int max = 0;
        int[][] dp = new int[m][n];

        // 第一列赋值
        for (int i = 0; i < m; i++) {
            dp[i][0] = matrix[i][0];
            max = Math.max(max, dp[i][0]);
        }

        // 第一行赋值
        for (int i = 0; i < n; i++) {
            dp[0][i] = matrix[0][i];
            max = Math.max(max, dp[0][i]);
        }

        // 递推
        for (int i = 1; i < m; i++) {
            for (int j = 1; j < n; j++) {
                dp[i][j] = matrix[i][j] == 1 ? Math.min(dp[i - 1][j - 1], Math.min(dp[i - 1][j], dp[i][j - 1])) + 1 : 0;

                max = Math.max(max, dp[i][j]);
            }
        }
        return max * max;
    }
}
```

### LeetCode version (`char[][]` as input) - (12ms 30.09%)

```java
class Solution {
    public int maximalSquare(char[][] matrix) {
        if (matrix == null || matrix.length == 0 || matrix[0].length == 0) {
            return 0;
        }
        int m = matrix.length;
        int n = matrix[0].length;

        int[][] dp = new int[m][n];
        int maxSqLen = 0;
        for (int i = 0; i < m; i++) {
            dp[i][0] = matrix[i][0] == '1' ? 1 : 0;
            maxSqLen = Math.max(dp[i][0], maxSqLen);
        }
        for (int j = 0; j < n; j++) {
            dp[0][j] = matrix[0][j] == '1' ? 1 : 0;
            maxSqLen = Math.max(dp[0][j], maxSqLen);
        }
        for (int i = 1; i < m; i++) {
            for (int j = 1; j < n; j++) {
                if (matrix[i][j] == '1') {
                    dp[i][j] = Math.min(Math.min(dp[i - 1][j], dp[i][j - 1]), dp[i - 1][j - 1]) + 1;
                    maxSqLen = Math.max(dp[i][j], maxSqLen);
                } 
            }
        }
        return maxSqLen * maxSqLen;
    }
}
```

### A smart way to bypass initialization, use `m + 1`, `n + 1` as `dp[][]` dimension:

(notice `matrix[i-1][j-1]` is used for determining `dp[i][j]`) - `O(mn)` time, `O(mn)` space - (13 ms, faster than 23.53%)

```java
public class Solution {
    public int maximalSquare(char[][] matrix) {
        int rows = matrix.length, cols = rows > 0 ? matrix[0].length : 0;
        int[][] dp = new int[rows + 1][cols + 1];
        int maxsqlen = 0;
        for (int i = 1; i <= rows; i++) {
            for (int j = 1; j <= cols; j++) {
                if (matrix[i-1][j-1] == '1'){
                    dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;
                    maxsqlen = Math.max(maxsqlen, dp[i][j]);
                }
            }
        }
        return maxsqlen * maxsqlen;
    }
}
```

### Space optimized DP - O(mn) time, O(n) space

```java
public class Solution {
    public int maximalSquare(char[][] matrix) {
        int rows = matrix.length, cols = rows > 0 ? matrix[0].length : 0;
        int[] dp = new int[cols + 1];
        int maxsqlen = 0, prev = 0;
        for (int i = 1; i <= rows; i++) {
            for (int j = 1; j <= cols; j++) {
                int temp = dp[j];
                if (matrix[i - 1][j - 1] == '1') {
                    dp[j] = Math.min(Math.min(dp[j - 1], prev), dp[j]) + 1;
                    maxsqlen = Math.max(maxsqlen, dp[j]);
                } else {
                    dp[j] = 0;
                }
                prev = temp;
            }
        }
        return maxsqlen * maxsqlen;
    }
}
```

## Reference

* [segmentfault: Maximal Square 最大正方形](https://segmentfault.com/a/1190000003709497)
* [LeetCode - Maximal Square](https://leetcode.com/articles/maximal-square/)