Leetcode #363: Max Sum of Rectangle No Larger Than K
In this guide, we solve Leetcode #363 Max Sum of Rectangle No Larger Than K in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Given an m x n matrix matrix and an integer k, return the max sum of a rectangle in the matrix such that its sum is no larger than k. It is guaranteed that there will be a rectangle with a sum no larger than k.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Array, Binary Search, Matrix, Ordered Set, Prefix Sum
Intuition
The problem structure suggests a monotonic decision, which makes binary search a natural fit.
By halving the search space each step, we reach the answer efficiently.
Approach
Search either directly on a sorted array or on the answer space using a check function.
Each check is fast, and the logarithmic search keeps the overall runtime low.
Steps:
- Define the search bounds.
- Check the mid point condition.
- Narrow the bounds until convergence.
Example
Input: matrix = [[1,0,1],[0,-2,3]], k = 2
Output: 2
Explanation: Because the sum of the blue rectangle [[0, 1], [-2, 3]] is 2, and 2 is the max number no larger than k (k = 2).
Python Solution
class Solution:
def maxSumSubmatrix(self, matrix: List[List[int]], k: int) -> int:
m, n = len(matrix), len(matrix[0])
ans = -inf
for i in range(m):
nums = [0] * n
for j in range(i, m):
for h in range(n):
nums[h] += matrix[j][h]
s = 0
ts = SortedSet([0])
for x in nums:
s += x
p = ts.bisect_left(s - k)
if p != len(ts):
ans = max(ans, s - ts[p])
ts.add(s)
return ans
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.