# Time:  ctor:   O(m * n)
#        lookup: O(1)
# Space: O(m * n)
#
# 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).
#
# Range Sum Query 2D
# 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:
# 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.

class NumMatrix(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 xrange(n+1)] for _ in xrange(m+1)]
        for i in xrange(1, m+1):
            for j in xrange(1, n+1):
                self.sums[i][j] = matrix[i-1][j-1]
                self.sums[i][j] += self.sums[i][j-1]
        for j in xrange(1, n+1):
            for i in xrange(1, m+1):
                self.sums[i][j] += self.sums[i-1][j]

    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]
        

# Your NumMatrix object will be instantiated and called as such:
# numMatrix = NumMatrix(matrix)
# numMatrix.sumRegion(0, 1, 2, 3)
# numMatrix.sumRegion(1, 2, 3, 4)

# Time:  ctor:   O(m * n)
#        lookup: O(1)
# Space: O(m * n)
#
# 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).
#
# Range Sum Query 2D
# 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:
# 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.

class NumMatrix(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 xrange(n+1)] for _ in xrange(m+1)]
        for i in xrange(1, m+1):
            for j in xrange(1, n+1):
                self.sums[i][j] = self.sums[i][j-1] + matrix[i-1][j-1]
        for j in xrange(1, n+1):
            for i in xrange(1, m+1):
                self.sums[i][j] += self.sums[i-1][j]

    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]
        

# Your NumMatrix object will be instantiated and called as such:
# numMatrix = NumMatrix(matrix)
# numMatrix.sumRegion(0, 1, 2, 3)
# numMatrix.sumRegion(1, 2, 3, 4)

Update range-sum-query-2d-immutable.py