Range sum query 2D mutable [BIT]

Time: O(MxN); Space: O(MxN); medium

Notes:

  • Time:

    • ctor: O(M * N)

    • update: O(LogM * LogN)

    • query: O(LogM * LogN)

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: 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]
]

Output:

  • sumRegion(2, 1, 4, 3) -> 8

  • update(3, 2, 2)

  • sumRegion(2, 1, 4, 3) -> 10

Notes:

  • The matrix is only modifiable by the update function.

  • You may assume the number of calls to update and sumRegion function is distributed evenly.

  • You may assume that row1 <= row2 and col1 <= col2.

1. Binary Indexed Tree (BIT) solution

[1]:

class NumMatrix(object):
    def __init__(self, matrix):
        """
        :type matrix: List[List[int]]
        """
        if not matrix:
            return
        self.__matrix = matrix
        self.__bit = [[0] * (len(self.__matrix[0]) + 1) \
                      for _ in range(len(self.__matrix) + 1)]
        for i in range(1, len(self.__bit)):
            for j in range(1, len(self.__bit[0])):
                self.__bit[i][j] = matrix[i-1][j-1] + self.__bit[i-1][j] + \
                                   self.__bit[i][j-1] - self.__bit[i-1][j-1]
        for i in reversed(range(1, len(self.__bit))):
            for j in reversed(range(1, len(self.__bit[0]))):
                last_i, last_j = i - (i & -i), j - (j & -j)
                self.__bit[i][j] = self.__bit[i][j] - self.__bit[i][last_j] - \
                                   self.__bit[last_i][j] + self.__bit[last_i][last_j]

    def update(self, row, col, val):
        """
        update the element at matrix[row,col] to val.
        :type row: int
        :type col: int
        :type val: int
        :rtype: void
        """
        if val - self.__matrix[row][col]:
            self.__add(row, col, val - self.__matrix[row][col])
            self.__matrix[row][col] = val

    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.__sum(row2, col2) - self.__sum(row2, col1 - 1) - \
               self.__sum(row1 - 1, col2) + self.__sum(row1 - 1, col1 - 1)

    def __sum(self, row, col):
        row += 1
        col += 1
        ret = 0
        i = row
        while i > 0:
            j = col
            while j > 0:
                ret += self.__bit[i][j]
                j -= (j & -j)
            i -= (i & -i)
        return ret

    def __add(self, row, col, val):
        row += 1
        col += 1
        i = row
        while i <= len(self.__matrix):
            j = col
            while j <= len(self.__matrix[0]):
                self.__bit[i][j] += val
                j += (j & -j)
            i += (i & -i)

[2]:

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]
]

numMatrix = NumMatrix(matrix)
assert numMatrix.sumRegion(2, 1, 4, 3) == 8

numMatrix.update(3, 2, 2)
assert numMatrix.sumRegion(2, 1, 4, 3) == 10