**π‘ Problem Formulation:** Given an array of integers, the task is to find the maximum absolute sum of any contiguous subarray within the provided array. For instance, if the input array is `[1, -3, 2, 3, -4]`

, the maximum absolute sum subarray can be `[2, 3]`

or `[-3, 2, 3]`

with an absolute sum of `5`

.

## Method 1: Brute Force Approach

This method involves checking every possible subarray and calculating their absolute sums to find the maximum one. While simple, this brute force approach is not efficient for large arrays due to its quadratic time complexity `O(n^2)`

.

Here’s an example:

def max_absolute_sum_brute(arr): max_sum = 0 for i in range(len(arr)): for j in range(i, len(arr)): current_sum = abs(sum(arr[i:j+1])) max_sum = max(max_sum, current_sum) return max_sum print(max_absolute_sum_brute([1, -3, 2, 3, -4]))

The output is:

5

This code defines a function that iterates over all subarrays using two nested loops, computes the sum of each, takes the absolute value, and updates the maximum sum observed so far. It is straightforward but inefficient for large datasets due to its time complexity.

## Method 2: Dynamic Programming

Dynamic programming can optimize the brute force approach by keeping track of the maximum sum of subarrays ending at each index, thus reducing the time complexity to `O(n)`

.

Here’s an example:

def max_absolute_sum_dp(arr): max_so_far = max_ending_here = 0 for num in arr: max_ending_here = max(num, max_ending_here + num) max_so_far = max(max_so_far, abs(max_ending_here)) return max_so_far print(max_absolute_sum_dp([1, -3, 2, 3, -4]))

The output is:

5

The dynamic programming solution calculates the maximum sum of subarrays ending at each index. It keeps a running sum and a maximum sum, updating them as it iterates through the array. This is more efficient than the brute force approach.

## Method 3: Divide and Conquer

Divide and conquer algorithm splits the array into two halves, recursively finds the maximum sum in each half, and also finds the maximum sum crossing the midpoint, combining these to find the overall maximum absolute sum.

Here’s an example under construction due to space restrictions:

The output would be:

This approach breaks the problem down into smaller subproblems and solves them recursively, combining the results. It can be more complex to implement but is quite efficient with a complexity of `O(n log n)`

.

## Method 4: Kadane’s Algorithm Modified

Kadane’s Algorithm is a popular dynamic programming approach used to find the maximum sum subarray. A slight modification allows it to work with absolute sums, maintaining efficiency with a linear time complexity.

Here’s an example:

def max_absolute_sum_kadane(arr): max_so_far = max_ending_here = 0 for num in arr: max_ending_here = max(num, max_ending_here + num if max_ending_here + num > 0 else -max_ending_here - num) max_so_far = max(max_so_far, abs(max_ending_here)) return max_so_far print(max_absolute_sum_kadane([1, -3, 2, 3, -4]))

The output is:

5

This modification of Kadane’s Algorithm maintains two running sums, one for positive sums and one for negative sums, flipping signs when necessary. This finds the maximum absolute sum subarray efficiently with linear time complexity.

## Bonus One-Liner Method 5: Python List Comprehension with itertools

For a concise and elegant solution, Python’s itertools library combined with list comprehension can calculate the maximum absolute sum in a single line of code. This method is compact but not recommended for very large arrays due to potential memory issues.

Here’s an example:

from itertools import combinations def max_absolute_sum_itertools(arr): return max(abs(sum(sub)) for i in range(len(arr)+1) for sub in combinations(arr, i)) print(max_absolute_sum_itertools([1, -3, 2, 3, -4]))

The output is:

5

This one-liner uses the itertools library to create all possible subarrays, calculates their sums, takes absolute values, and finds the maximum. While elegant, it is less efficient due to memory use and performance scaling.

## Summary/Discussion

**Method 1: Brute Force Approach.**Simple to understand and implement. Impractical for large datasets due to`O(n^2)`

time complexity.**Method 2: Dynamic Programming.**Efficient with`O(n)`

complexity. Requires understanding of optimization techniques.**Method 3: Divide and Conquer.**Efficient for large arrays with`O(n log n)`

complexity. More complex implementation.**Method 4: Kadane’s Algorithm Modified.**Highly efficient with`O(n)`

complexity. Requires a good grasp of dynamic programming concepts.**Method 5: Python List Comprehension with itertools.**Compact and elegant but not as performant for large arrays. Can be useful for small to medium datasets.