# Question

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 row1 ≤ row2 and col1 ≤ col2.

# Solution

class NumMatrix(object):

def __init__(self, matrix):
"""
:type matrix: List[List[int]]
"""
if len(matrix) == 0 or len(matrix[0]) == 0:
self.X, self.Y = 0, 0
self.sum = [[]]
return

self.X = len(matrix)
self.Y = len(matrix[0])
self.sum = list()
self.sum.append([matrix[0][0]])
for i in range(1, self.Y):
self.sum[0].append(self.sum[0][-1] + matrix[0][i])

for i in range(1, self.X):
prev = matrix[i][0]
self.sum.append([self.sum[i - 1][0] + matrix[i][0]])
for j in range(1, self.Y):
prev += matrix[i][j]
self.sum[i].append(self.sum[i - 1][j] + prev)

def sumRegion(self, row1, col1, row2, col2):
"""
:type row1: int
:type col1: int
:type row2: int
:type col2: int
:rtype: int
"""
return self.sum[row2][col2] - \
(self.sum[row2][col1 - 1] if col1 > 0 else 0) - \
(self.sum[row1 - 1][col2] if row1 > 0 else 0) + \
(self.sum[row1 - 1][col1 - 1] if col1 > 0 and row1 > 0 else 0)

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