Maximum Side Length of a Square with Sum Less than or Equal to Threshold

Maximum Side Length of a Square with Sum Less than or Equal to Threshold

1. What is this problem about?

This coding problem asks you to find the maximum side length k of a square subgrid such that the sum of all elements within that k x k square is less than or equal to a given threshold. You are given a 2D matrix of integers and the threshold value.

2. Why is this asked in interviews?

DE Shaw and Amazon ask this problem to test a candidate's ability to combine multiple optimization techniques: 2D Prefix Sums and Binary Search. It requires you to precalculate the grid's sums to allow for O(1) area queries, making it a great test of your spatial reasoning and complexity analysis skills.

3. Algorithmic pattern used

This utilizes the Matrix, Binary Search, and Prefix Sum interview pattern. First, create a 2D Prefix Sum matrix where pref[i][j] is the sum of the subgrid from (0,0) to (i,j). Then, use binary search on the possible side lengths k (from 1 to min(rows, cols)). For each k, iterate through all possible top-left corners of a k x k square and use the prefix sum formula to check the square's sum in O(1) time.

4. Example explanation

2x2 Matrix: [[1, 1], [1, 1]], Threshold: 3.

• Check k = 1: Any 1x1 square sum is 1. (Valid, 1 <= 3).
• Check k = 2: The only 2x2 square sum is 1+1+1+1 = 4. (Invalid, 4 > 3). The maximum side length is 1.

5. Common mistakes candidates make

A frequent error is incorrectly implementing the 2D prefix sum formula, which is Total = Pref[r2][c2] - Pref[r1-1][c2] - Pref[r2][c1-1] + Pref[r1-1][c1-1]. Another mistake is not using binary search, resulting in a slower O(NMmin(N,M)) solution instead of O(NMlog(min(N,M))).

6. Interview preparation tip

For "Matrix, Prefix Sum interview pattern" problems, always draw a small grid and derive the formula for a subgrid sum yourself. It's easy to make off-by-one errors with indices, so careful visualization is your best defense.