To solve this problem, use binary search on the sum of possible rectangle sums and apply prefix sum optimization. The goal is to maximize the rectangle sum while ensuring it's no larger than k. By applying an ordered set to track valid sums, you can efficiently find the best solution in a matrix.
Problem Statement
You are given an m x n matrix, along with an integer k. Your task is to find the maximum sum of any rectangle within the matrix such that the sum does not exceed k. It is guaranteed that a valid rectangle will exist with a sum no larger than k.
For instance, in a matrix with the values [[1, 0, 1], [0, -2, 3]] and k = 2, the rectangle with the sum of 2 is valid. You are required to implement an efficient solution that handles both space and time constraints effectively.
Examples
Example 1
Input: matrix = [[1,0,1],[0,-2,3]], k = 2
Output: 2
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).
Example 2
Input: matrix = [[2,2,-1]], k = 3
Output: 3
Example details omitted.
Constraints
- m == matrix.length
- n == matrix[i].length
- 1 <= m, n <= 100
- -100 <= matrix[i][j] <= 100
- -105 <= k <= 105
Solution Approach
Binary Search on Sum Space
The core approach involves performing binary search over the possible sum values of the rectangle, adjusting the search space based on the sum constraints. This search helps narrow down the optimal sum that is less than or equal to k.
Prefix Sum Optimization
Prefix sum arrays are used to quickly calculate the sum of submatrices. This allows for efficient evaluation of different rectangle sums without recalculating values repeatedly.
Ordered Set for Efficient Sum Tracking
By maintaining an ordered set of possible sums for the rectangles, the solution efficiently tracks the largest possible valid sum no larger than k, reducing redundant computations.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on the binary search and matrix operations. In the worst case, the solution may involve iterating through rows and columns, leading to a time complexity of O(n^2 log k). Space complexity is influenced by the need to store prefix sums and ordered sets, resulting in O(n^2) space.
What Interviewers Usually Probe
- Ability to efficiently handle a dynamic programming problem with constraints.
- Skill in optimizing matrix-based solutions with space and time complexity considerations.
- Proficiency in using binary search and ordered data structures for algorithmic optimization.
Common Pitfalls or Variants
Common pitfalls
- Failing to correctly implement the binary search to narrow down valid sums.
- Neglecting to optimize sum calculations using prefix sums, leading to inefficient solutions.
- Improper handling of edge cases, such as when k is very small or the matrix contains negative values.
Follow-up variants
- Max Sum of Submatrix with Exact Sum: Instead of maximizing the sum, the goal is to find a submatrix with a sum equal to k.
- Max Sum of Rectangle No Larger Than K with Multiple Queries: Handling multiple k values for the same matrix and optimizing for each query.
- Max Sum of Subrectangle in a 2D Matrix with Larger k: An extension where k is larger, requiring different search space handling.
How GhostInterview Helps
- GhostInterview provides an in-depth breakdown of solving the Max Sum of Rectangle No Larger Than K problem using binary search and prefix sums, ensuring interview-ready performance.
- It assists in identifying key optimizations, like ordered sets and binary search, to improve the efficiency of the solution for both time and space complexities.
- By explaining common pitfalls and offering practice problems based on this pattern, GhostInterview prepares you to tackle similar matrix-based dynamic programming questions.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the optimal approach for solving Max Sum of Rectangle No Larger Than K?
The optimal approach involves binary search on the sum space combined with prefix sum arrays and ordered sets to efficiently track the maximum valid rectangle sum.
How can I optimize the space complexity for this problem?
Using prefix sums and ordered sets helps in minimizing redundant space usage, allowing for efficient tracking of sums without recalculating them multiple times.
What is the time complexity of solving the Max Sum of Rectangle No Larger Than K problem?
The time complexity depends on the matrix size and the binary search, which typically leads to O(n^2 log k) complexity in the worst case.
What are common mistakes when solving this problem?
Common mistakes include failing to implement binary search correctly, neglecting to optimize with prefix sums, or improperly handling edge cases like negative values in the matrix.
How does binary search improve the solution to this problem?
Binary search helps by narrowing the possible sum space, reducing the number of potential rectangle sums that need to be checked, which optimizes performance.
Need direct help with Max Sum of Rectangle No Larger Than K instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Max Sum of Rectangle No Larger Than K from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Design an algorithm to pick random points within non-overlapping rectangles using binary search and reservoir sampling.
Open problem page#731 My Calendar IIImplement a calendar that allows double bookings but prevents triple bookings, managing overlapping intervals efficiently with binary search.
Open problem page#1292 Maximum Side Length of a Square with Sum Less than or Equal to ThresholdThis problem challenges you to find the largest square in a matrix with a sum below a given threshold using binary search and prefix sums.
Open problem page