**π‘ Problem Formulation:** In computational programming, a common task is to find a contiguous subarray within a one-dimensional array or list that has the largest sum, given a fixed length. For example, given an array `[1, 2, 3, 4, 5]`

and a subarray length of `2`

, the desired output would be `[4, 5]`

, as it has the largest sum (9) among all subarrays of length 2.

## Method 1: Brute Force

The brute force method iterates over every possible subarray of the given length and calculates its sum, keeping track of the largest sum and its corresponding subarray. Although this method is simple to implement, it is not efficient for large arrays due to its O(n*k) time complexity, where n is the length of the array and k is the subarray length.

Here’s an example:

def max_subarray(nums, k): max_sum = float('-inf') max_subarray = [] for i in range(len(nums) - k + 1): current_sum = sum(nums[i:i+k]) if current_sum > max_sum: max_sum = current_sum max_subarray = nums[i:i+k] return max_subarray print(max_subarray([1, 2, 3, 4, 5], 2))

Output: `[4, 5]`

This code snippet uses a for loop to walk through the array, calculates the sum of each possible subarray and updates the maximum value when a larger sum is found. The largest subarray is then returned as output.

## Method 2: Sliding Window

The sliding window technique optimizes the brute force approach by eliminating the need to calculate the sum of each subarray from scratch. It uses a window to slide across the original array and updates the sum by subtracting the element going out of the window and adding the new element coming into the window. This approach runs in O(n) time, significantly faster for larger arrays.

Here’s an example:

def max_subarray(nums, k): current_sum = max_sum = sum(nums[:k]) max_subarray_start = 0 for i in range(k, len(nums)): current_sum += nums[i] - nums[i-k] if current_sum > max_sum: max_sum = current_sum max_subarray_start = i - k + 1 return nums[max_subarray_start:max_subarray_start+k] print(max_subarray([1, 2, 3, 4, 5], 2))

Output: `[4, 5]`

This code initializes the sum with the first k elements and iteratively adjusts the sum as it slides over the array, tracking the largest sum and starting index of the maximum subarray.

## Method 3: Dynamic Programming

Dynamic programming is used to keep track of sums in an array where each index corresponds to the largest sum possible up to that point. By avoiding re-calculation of sums, this method reduces redundant work. However, dynamic programming is not the most optimal solution for fixed-length subarray problems. It is better suited for variable-length subarrays.

Unfortunately, there is no direct application of dynamic programming for the fixed-length subarray problem that isn’t better solved by the sliding window technique. In the interest of staying true to the problem, we suggest sticking with the sliding window method for this specific use case and reserve dynamic programming for variations where it’s more applicable.

## Method 4: Divide and Conquer

This method splits the array into two halves and finds the maximum subarray within each half and the crossing subarray which overlaps the two halves. The largest of these is the solution. Though this is a powerful algorithmic paradigm, for finding fixed-length subarrays, it’s more complex than necessary.

In the context of fixed-length subarrays, divide and conquer is less practical compared to the sliding window method due to its unnecessary complexity and lack of substantial performance gain for the problem at hand.

## Bonus One-Liner Method 5: Pythonic Way with max() and List Comprehension

Python’s `max()`

function combined with list comprehension offers a succinct one-liner to tackle this problem. Despite the elegance, this method still has an O(n*k) complexity, similar to the brute force approach.

Here’s an example:

nums = [1, 2, 3, 4, 5] k = 2 print(max((nums[i:i+k] for i in range(len(nums)-k+1)), key=sum))

Output: `[4, 5]`

This code uses list comprehension to create every subarray and utilizes the `max()`

function to find the one with the largest sum. It’s compact and takes advantage of Python’s comprehensive standard library.

## Summary/Discussion

**Method 1: Brute Force.**Simple to understand and implement. Inefficient for larger data sets.**Method 2: Sliding Window.**Efficient and optimal with O(n) complexity. Best for large arrays.**Method 3: Dynamic Programming.**Not ideally suited for fixed-length subarray problems; better for variable-length subarray optimization.**Method 4: Divide and Conquer.**Powerful but unnecessarily complex for fixed-length subarrays. Not the most practical solution.**Bonus Method 5: Pythonic Way.**Elegant one-liner. However, it’s no more efficient than the brute force method.