Range Sum Query 2D - Immutable

Given a 2D matrix matrix, find the sum of the elements inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2).

The above rectangle (with the red border) is defined by (row1, col1) = (2, 1) and (row2, col2) = (4, 3), which contains sum = 8.

Example:

Given matrix = [
  [3, 0, 1, 4, 2],
  [5, 6, 3, 2, 1],
  [1, 2, 0, 1, 5],
  [4, 1, 0, 1, 7],
  [1, 0, 3, 0, 5]
]

sumRegion(2, 1, 4, 3) -> 8
sumRegion(1, 1, 2, 2) -> 11
sumRegion(1, 2, 2, 4) -> 12

Note:

1. You may assume that the matrix does not change.

2. There are many calls to sumRegion function.

3. You may assume that row1 ≤ row2 and col1 ≤ col2.

这题在前一题基础上，扩展成2d的问题。思路还是一样，用类似dp的方法算 prefixsum[i][j]

思路如下，在leetcode discuss抄来的，懒得自己写了。

计算 prefix sum array:

+-----+-+-------+     +--------+-----+     +-----+---------+     +-----+--------+     +---------------+ 
|     | |       |     |        |     |     |     |         |     |     |        |     |               |
|     | |       |     |        |     |     |     |         |     |     |        |     |               |
+-----+-+       |     +--------+     |     |     |         |     +-----+        |     |     +-+       | 
|     | |       |  =  |              |  +  |     |         |  -  |              |  +  |     | |       |
+-----+-+       |     |              |     +-----+         |     |              |     |     +-+       |
|               |     |              |     |               |     |              |     |               |
|               |     |              |     |               |     |              |     |               |
+---------------+     +--------------+     +---------------+     +--------------+     +---------------+

   sums[i][j]      =    sums[i-1][j]    +     sums[i][j-1]    -   sums[i-1][j-1]   +      matrix[i-1][j-1]

取值：

+---------------+   +--------------+   +---------------+   +--------------+   +--------------+
|               |   |         |    |   |   |           |   |         |    |   |   |          |
|   (r1,c1)     |   |         |    |   |   |           |   |         |    |   |   |          |
|   +------+    |   |         |    |   |   |           |   +---------+    |   +---+          |
|   |      |    | = |         |    | - |   |           | - |      (r1,c2) | + |   (r1,c1)    |
|   |      |    |   |         |    |   |   |           |   |              |   |              |
|   +------+    |   +---------+    |   +---+           |   |              |   |              |
|        (r2,c2)|   |       (r2,c2)|   |   (r2,c1)     |   |              |   |              |
+---------------+   +--------------+   +---------------+   +--------------+   +--------------+

程序如下

class NumMatrix(object):

    def __init__(self, matrix):
        """
        :type matrix: List[List[int]]
        """
        if len(matrix) == 0 or len(matrix[0]) == 0:
            m = n = 0
        else:
            m, n = len(matrix), len(matrix[0])
        self.prefix = [[0] * (n + 1) for _ in xrange(m + 1)]
        for i in xrange(1, m + 1):
            for j in xrange(1, n + 1):
                self.prefix[i][j] = self.prefix[i-1][j] + self.prefix[i][j-1] \
                                    - self.prefix[i-1][j-1] + matrix[i-1][j-1]


    def sumRegion(self, row1, col1, row2, col2):
        """
        :type row1: int
        :type col1: int
        :type row2: int
        :type col2: int
        :rtype: int
        """
        return self.prefix[row2 + 1][col2 + 1] + self.prefix[row1][col1] \
               - self.prefix[row1][col2 + 1] - self.prefix[row2 + 1][col1]