max_sum_of_rectangle_no_larger_than_k

Problem

Given a non-empty 2D matrix matrix and an integer k, find the max sum of a rectangle in the matrix such that its sum is no larger than k.

Example:

Given matrix = [
  [1,  0, 1],
  [0, -2, 3],
]
k = 2

The answer is 2. Because the sum of rectangle [[0, 1], [-2, 3]] is 2 and 2 is the max number no larger than k (k = 2).

Note:

• The rectangle inside the matrix must have an area > 0.
• What if the number of rows is much larger than the number of columns?

Solution

• Use cumulative sums in 1 dimension
• Max sum close to number k can be done in O(n logn)