Range sum query 2D - immutable

Time: ctor:O(MxN), lookup:O(1); Space: O(MxN); medium

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 1:

Input/Output: 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

Explanation:

  • sumRegion(2, 1, 4, 3) = 2 + 0 + 1 + 1 + 0 + 1 + 0 + 3 + 0 = 8

  • sumRegion(1, 1, 2, 2) = 6 + 3 + 2 + 0 = 11

  • sumRegion(1, 2, 2, 4) = 3 + 2 + 1 + 0 + 1 + 5 = 12

Example 2:

Input/Output: matrix =

[
    [3,0],
    [5,6]
]

  • sumRegion(0, 0, 0, 1) -> 3

  • sumRegion(0, 0, 1, 1) -> 14

Explanation:

  • sumRegion(0, 0, 0, 1) = 3 + 0 = 3

  • sumRegion(0, 0, 1, 1) = 3 + 0 + 5 + 6 = 14

Notes:

  • You may assume that the matrix does not change.

  • There are many calls to sumRegion function.

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

[3]:

class NumMatrix1(object):

    def __init__(self, matrix):
        """
        initialize your data structure here.
        :type matrix: List[List[int]]
        """
        if not matrix:
            return

        m, n = len(matrix), len(matrix[0])
        self.__sums = [[0 for _ in range(n+1)] for _ in range(m+1)]

        for i in range(1, m + 1):
            for j in range(1, n + 1):
                self.__sums[i][j] = self.__sums[i][j-1] + self.__sums[i-1][j] - \
                                    self.__sums[i-1][j-1] + matrix[i-1][j-1]

    def sumRegion(self, row1, col1, row2, col2):
        """
        sum of elements matrix[(row1,col1)..(row2,col2)], inclusive.
        :type row1: int
        :type col1: int
        :type row2: int
        :type col2: int
        :rtype: int
        """
        return self.__sums[row2+1][col2+1] - self.__sums[row2+1][col1] - \
               self.__sums[row1][col2+1] + self.__sums[row1][col1]

[5]:

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]
]
n = NumMatrix1(matrix)
assert n.sumRegion(2, 1, 4, 3) == 8
assert n.sumRegion(1, 1, 2, 2) == 11
assert n.sumRegion(1, 2, 2, 4) == 12

matrix = [
  [3,0],
  [5,6]
]
n = NumMatrix1(matrix)
assert n.sumRegion(0, 0, 0, 1) == 3
assert n.sumRegion(0, 0, 1, 1) == 14