old_contents: # 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)
  
new_contents: # 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)
  
subject: Update range-sum-query-2d-immutable.py