Sliding Window Matrix Maximum

Example 1:

Input：[[1,5,3],[3,2,1],[4,1,9]]，k=2
Output：13
Explanation：
At first the window is at the start of the matrix like this

	[
	  [|1, 5|, 3],
	  [|3, 2|, 1],
	  [4, 1, 9],
	]
,get the sum 11;
then the window move one step forward.

	[
	  [1, |5, 3|],
	  [3, |2, 1|],
	  [4, 1, 9],
	]
,get the sum 11;
then the window move one step forward again.

	[
	  [1, 5, 3],
	  [|3, 2|, 1],
	  [|4, 1|, 9],
	]
,get the sum 10;
then the window move one step forward again.

	[
	  [1, 5, 3],
	  [3, |2, 1|],
	  [4, |1, 9|],
	]
,get the sum 13;
SO finally, get the maximum from all the sum which is 13.

Example 2:

Input：[[10]，k=1
Output：10
Explanation：
sliding window size is 1*1，and return 10.

O(n^2) time.

思路：

考点：

• 二维前缀和

题解：

• sum[i][j]存储左上角坐标为(0,0),右下角坐标为(i,j)的子矩阵的和。
• sum[i][j] = matrix[i - 1][j - 1] + sum[i - 1][j] + sum[i][j - 1] - sum[i - 1][j - 1];递推求值即可，两部分相加，减去重复计算部分。
• int value = sum[i][j] - sum[i - k][j] -sum[i][j - k] + sum[i - k][j - k];可求得一个k * k大小子矩阵的和。

  public class Solution {
      /**
       * @param matrix: an integer array of n * m matrix
       * @param k: An integer
       * @return: the maximum number
       */
     public int maxSlidingMatrix(int[][] matrix, int k) {
          // Write your code here
          int n = matrix.length;
          if (n == 0 || n < k)
              return 0;
          int m = matrix[0].length;
          if (m == 0 || m < k)
              return 0;
  
          int[][] sum = new int[n + 1][m + 1];
          for (int i = 0; i <= n; ++i) sum[i][0] = 0;
          for (int i = 0; i <= m; ++i) sum[0][i] = 0;
  
          for (int i = 1; i <= n; ++i)
              for (int j = 1; j <= m; ++j)
                  sum[i][j] = matrix[i - 1][j - 1] + 
                              sum[i - 1][j] + sum[i][j - 1] - sum[i - 1][j - 1];
  
          int max_value = Integer.MIN_VALUE;
          for (int i = k; i <= n; ++i)
              for (int j = k; j <= m; ++j) {
                  int value = sum[i][j] - sum[i - k][j] -
                              sum[i][j - k] + sum[i - k][j - k];
  
                  if (value > max_value)
                      max_value = value;
              }
          return max_value;
      }
  }