What is the optimal time complexity of solving Maximum Sum Rectangle?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Maximum Sum Rectangle?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Maximum Sum Rectangle?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Maximum Sum Rectangle?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingHard

Maximum Sum Rectangle

Learn how to solve the 'Maximum Sum Rectangle' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Hard

Write a function max_sum_rectangle(matrix) that finds and returns the maximum sum of a sub-matrix in a 2D matrix of integers.

Constraints

•1 <= len(matrix), len(matrix[0]) <= 100
•-1000 <= matrix[i][j] <= 1000

Examples

Example 1

Input

max_sum_rectangle([[1, 2, -1, -4, -20], [-8, -3, 4, 2, 1], [3, 8, 10, 1, 3], [-4, -1, 1, 7, -6]])

Output

Explanation

The sub-matrix from row 1, column 1 to row 3, column 3 has the maximum sum of 29.

Example 2

Input

max_sum_rectangle([[-1, -2], [-3, -4]])

Output

-1

Explanation

The maximum sum is -1, which is the cell at (0, 0).

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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