304.Range Sum Query 2D - Immutable

Tags: [dp], [two_dimension_dp], [preprocessing]

Link: https://leetcode.com/problems/range-sum-query-2d-immutable/#/description

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 row 1 ≤ row2 and col1 ≤ col2.

Solution: DP, Two Dimension DP

class NumMatrix(object):

    def __init__(self, matrix):
        """
        :type matrix: List[List[int]]
        """
        self.matrix_is_empty = False
        if not matrix or not matrix[0]:
            self.matrix_is_empty = True
            return

        self.num_of_rows = len(matrix)
        self.num_of_cols = len(matrix[0])

        # sum_memo[r][c] is recording the sum of the submatrix,
        # whose left top corner is [r][c] and right bottom corner is [num_of_rows - 1][num_of_cols - 1].
        self.sum_memo = [[0 for _ in xrange(self.num_of_cols)] for _ in xrange(self.num_of_rows)]
        self.sum_memo[self.num_of_rows - 1][self.num_of_cols - 1] = matrix[self.num_of_rows - 1][self.num_of_cols - 1]

        for col in xrange(self.num_of_cols - 2, -1, -1):
            self.sum_memo[self.num_of_rows - 1][col] = matrix[self.num_of_rows - 1][col] +\
                                                       self.sum_memo[self.num_of_rows - 1][col + 1];

        for row in xrange(self.num_of_rows - 2, -1, -1):
            self.sum_memo[row][self.num_of_cols - 1] = matrix[row][self.num_of_cols - 1] +\
                                                       self.sum_memo[row + 1][self.num_of_cols - 1]

        for row  in xrange(self.num_of_rows - 2, -1, -1):
            for col in xrange(self.num_of_cols - 2, -1, -1):
                self.sum_memo[row][col] = matrix[row][col] +\
                                          self.sum_memo[row][col + 1] +\
                                          self.sum_memo[row + 1][col] -\
                                          self.sum_memo[row + 1][col + 1]

    def sumRegion(self, row1, col1, row2, col2):
        """
        :type row1: int
        :type col1: int
        :type row2: int
        :type col2: int
        :rtype: int
        """
        if self.matrix_is_empty:
            return 0
        if not (0 <= row1 <= self.num_of_rows) or not (0 <= col1 <= self.num_of_cols) or\
           not (0 <= row2 <= self.num_of_rows) or not (0 <= col2 <= self.num_of_cols):
               return 0

        row_left_bottom = row2
        col_left_bottom = col1

        row_right_top = row1
        col_right_top = col2

        result = self.sum_memo[row1][col1]
        if row_left_bottom + 1 < self.num_of_rows:
            result -= self.sum_memo[row_left_bottom + 1][col_left_bottom]
        if col_right_top + 1 < self.num_of_cols:
            result -= self.sum_memo[row_right_top][col_right_top + 1]
        if row2 + 1 < self.num_of_rows and col2 + 1 < self.num_of_cols:
            result += self.sum_memo[row2 + 1][col2 + 1]

        return result

# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# param_1 = obj.sumRegion(row1,col1,row2,col2)

Note:

• Time complexity of initialization = O(num_of_rows * num_of_cols).
• Time complexity of sumRange = O(1).