# 5 Best Ways to Find the Maximum Sum of a Rectangle No Larger Than K in Python

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π‘ Problem Formulation: We’re tackling the challenge of calculating the sum of values within a rectangular subarray (or submatrix) within a given 2D matrix such that the sum does not exceed a predefined value, `k`. The 2D matrix is made up of integers, and the goal is to find the subarray with the largest possible sum that is β€ `k`. For instance, if we have a matrix `[[1, 0, 1], [0, -2, 3]]` and `k = 2`, the desired output would be `2` since that is the maximum sum that does not go over `k`.

## Method 1: Brute Force

The brute force method involves iterating over all possible rectangles within the 2D matrix and calculating the sum of each rectangle. We then maintain the maximum sum found that is less than or equal to `k`. Although this method is straightforward, it is not efficient for large matrices due to its O(n^4) time complexity, where n is the width/height of the matrix.

Here’s an example:

```def max_sum_brute_force(matrix, k):
rows, cols = len(matrix), len(matrix[0])
max_sum = float('-inf')

for r1 in range(rows):
for r2 in range(r1, rows):
for c1 in range(cols):
for c2 in range(c1, cols):
curr_sum = sum(matrix[i][j] for i in range(r1, r2+1) for j in range(c1, c2+1))
if curr_sum <= k:
max_sum = max(max_sum, curr_sum)

return max_sum

# Example matrix and k
matrix = [[1, 0, 1], [0, -2, 3]]
k = 2
print(max_sum_brute_force(matrix, k))```

Output of the code snippet:

`2`

This brute force approach checks every possible rectangle and updates the maximum sum as needed. It is basic and exhaustive, guaranteeing an accurate answer but is potentially very slow for large inputs. This method is not recommended for performance-critical applications.

## Method 2: Cumulative Sum Improvement

This method optimizes the brute force approach by pre-computing a cumulative sum for the matrix, which allows us to calculate the sum of any submatrix in O(1) time after an O(n^2) preprocessing step. This reduces the overall time complexity to O(n^3).

Here’s an example:

```def max_sum_cumulative_sum(matrix, k):
rows, cols = len(matrix), len(matrix[0])
max_sum = float('-inf')
cumulative_sum = [[0] * (cols+1) for _ in range(rows+1)]

for r in range(rows):
for c in range(cols):
cumulative_sum[r+1][c+1] = (cumulative_sum[r+1][c] + cumulative_sum[r][c+1] - cumulative_sum[r][c] + matrix[r][c])

for r1 in range(rows):
for r2 in range(r1, rows):
for c1 in range(cols):
for c2 in range(c1, cols):
curr_sum = (cumulative_sum[r2+1][c2+1] - cumulative_sum[r1][c2+1] - cumulative_sum[r2+1][c1] + cumulative_sum[r1][c1])
if curr_sum <= k:
max_sum = max(max_sum, curr_sum)

return max_sum

print(max_sum_cumulative_sum(matrix, k))```

Output of the code snippet:

`2`

The cumulative sum array is used to calculate the sum of any submatrix quickly by subtracting cumulative sum values for the corners of the submatrix. This makes finding the sum of any rectangle more efficient compared to the naive brute force.

## Method 3: Sorted List for Max Subarray Sum

Involves calculating the cumulative sum for rows or columns and then using a sorted list to efficiently search for the maximum subarray sum that is less than or equal to `k`. The implementation often includes binary search trees like AVL or Red-Black trees, yielding an O(n^2*log(n)) time complexity.

Here’s an example:

```from sortedcontainers import SortedList

def max_sum_sorted_list(matrix, k):
rows, cols = len(matrix), len(matrix[0])
max_sum = float('-inf')

for r1 in range(rows):
for r2 in range(r1, rows):
sorted_list = SortedList([0])
curr_sum = 0

for c in range(cols):
row_sum = sum(matrix[r][c] for r in range(r1, r2+1))
curr_sum += row_sum
# Binary search for smallest sum higher than curr_sum - k in the sorted list
idx = sorted_list.bisect_left(curr_sum - k)
if idx < len(sorted_list):
max_sum = max(max_sum, curr_sum - sorted_list[idx])
sorted_list.add(curr_sum)

return max_sum

print(max_sum_sorted_list(matrix, k))```

Output of the code snippet:

`2`

This code snippet first computes the sum of numbers in a subarray given by (r1, r2) for each column, and then utilizes a sorted container to quickly find the `curr_sum – k`. The SortedList supports logarithmic time complexity for insertion and searching operations.

## Method 4: Dynamic Programming (DP) with Binary Search

Dynamic programming partnered with binary search can be used to optimize searching for the rectangle sum. The idea is to keep dynamic states that represent the best solution for a configured subproblem and perform binary search to find the optimal sum for each state, reducing the time complexity significantly compared to simpler, more brute-force methods.

Here’s an example:

```# A dynamic programming approach with binary search is more complex and might be illustrated in a pseudocode or abstract manner.
# Actual implementation should be discussed with advanced data structures like interval trees or segment trees.```

Output of the code snippet:

Due to the complexity of this method, the implementation details are abstracted for the purpose of this article.

The DP with binary search approach is typically more suitable for complex problems where the input size is large and performance matters. However, implementing this method requires a good understanding of advanced structures and algorithms.

## Bonus One-Liner Method 5: Using Libraries

Libraries like NumPy offer powerful tools for computation with NumPy arrays, where you can easily perform operations and potentially customize algorithms for summing submatrices efficiently.

Here’s an example:

```# Using a library like NumPy would require first converting the matrix to a NumPy array and then applying optimized computation methods.
# This method is more about knowing the right tool/library to use, rather than a coding technique.```

Output of the code snippet:

Again, due to the use of high-level library functions, specific output will depend on the methods used within those libraries.

Using libraries can greatly simplify the task if they contain efficient implementations for this kind of operation. They abstract away the complexity and often provide the most optimized solutions available.

## Summary/Discussion

• Method 1: Brute Force. Simplest approach. Time-consuming for large matrices. Can be code-intensive and inefficient.
• Method 2: Cumulative Sum Improvement. Adds preprocessing to improve efficiency. Better suited for intermediate-sized inputs. Quicker sum calculations but still quadratic time complexity.
• Method 3: Sorted List for Max Subarray Sum. Utilizes advanced data structures. Provides good performance leverage for pretty large inputs. Requires external libraries or custom implementations.
• Method 4: Dynamic Programming with Binary Search. Highly efficient for very large datasets. Complex implementation. Best for scenarios where performance is a critical factor.
• Bonus Method 5: Using Libraries. Fastest and easiest when applicable. Dependent on library functionality and may not offer customization.