Given a 2D matrixmatrix, 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.
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
The Idea: The general idea is to only accumulate on a row or column basis so that we only have to operate on the basis of either just rows or columns.
Complexity: O(n) time for both update and sum, and O(n) extra space where n is the number of columns
import numpy as np
class NumMatrix:
def __init__(self, matrix):
"""
initialize your data structure here.
:type matrix: List[List[int]]
"""
self.d = matrix
for row in matrix:
NumMatrix.cum_sum(row)
@staticmethod
def cum_sum(row):
for i in range(1, len(row)):
row[i] += row[i-1]
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
"""
# this is the old cdf
cdf = self.d[row]
# can build pdf from the cdf and properly insert value
prev = 0
pdf = []
for row_elm in cdf:
pdf.append(row_elm - prev)
prev = row_elm
pdf[col] = val
# now update the old pdf
for i in range(1, len(pdf)):
pdf[i] += pdf[i-1]
self.d[row] = pdf
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
"""
# integral[5, 10] in discrete -> F[10] - F[5-1]
# but treat this on a 2 dimensional scale
right_sum, left_sum = 0, 0
for i in range(row1, row2+1):
right_sum += self.d[i][col2]
if col1 - 1 >= 0:
for i in range(row1, row2 + 1):
left_sum += self.d[i][col1 - 1]
return right_sum - left_sum
# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# obj.update(row,col,val)
# param_2 = obj.sumRegion(row1,col1,row2,col2)
obj = NumMatrix([[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]])
print(obj.sumRegion(1,1,3,3))
print(obj.sumRegion(2,2,3,3))
print(obj.update(2,2,100))
print(obj.sumRegion(2,2,3,3))
