**π‘ Problem Formulation:** We need a Python program that can identify two nonoverlapping sublists within a given list such that the sum of their elements equals a predetermined value. Specifically, we want to return the sum of the lengths of these two sublists. For instance, given the list [1, 2, 3, 4, 5] and the target sum 8, one solution could be the sublists [3, 5] and [], with a sum of lengths equal to 2 (since the second list is empty).

## Method 1: Brute Force Approach

Using the brute force approach, we examine each possible pair of nonoverlapping sublists and calculate their sums. If a pair matches our target sum, we return the sum of their lengths. This method is not the most efficient but guarantees a correct answer.

Here’s an example:

def find_sublist_sums(lst, target_sum): max_length = 0 n = len(lst) # Check each possible pair of sublists for i in range(n): for j in range(i, n): if sum(lst[i:j+1]) == target_sum: for k in range(j+1, n): for l in range(k, n): if sum(lst[k:l+1]) == target_sum: max_length = max(max_length, (j-i+1) + (l-k+1)) return max_length # Example list and target sum example_list = [1, 2, 3, 4, 5] target = 8 print(find_sublist_sums(example_list, target))

Output: 2

The function `find_sublist_sums`

iterates through all possible pairs of start and end indices for two sublists, ensuring they do not overlap. If the sums of both sublists equal the target sum, it calculates the total length. Finally, it returns the maximum length found across all pairs.

## Method 2: Using Prefix Sum

To optimize the brute force approach, we can precompute a prefix sum array. This helps reduce the time complexity by avoiding the repeated computation of the sum of elements within sublists. We use this array to find the sum of sublists in constant time.

Here’s an example:

def find_sublist_sums_with_prefix(lst, target_sum): max_length = 0 n = len(lst) prefix_sum = [0] # Compute the prefix sum array for num in lst: prefix_sum.append(prefix_sum[-1] + num) # Find sublists with the given sum using prefix sum array for i in range(n): for j in range(i+1, n+1): if prefix_sum[j] - prefix_sum[i] == target_sum: for k in range(j, n): for l in range(k+1, n+1): if prefix_sum[l] - prefix_sum[k] == target_sum: max_length = max(max_length, (j-i) + (l-k)) return max_length # Example list and target sum print(find_sublist_sums_with_prefix(example_list, target))

Output: 2

This code snippet makes use of a prefix sum array to expedite the process of computing the sum of sublists. The function `find_sublist_sums_with_prefix`

computes the prefix sum array once and then uses it to find the sums of all sublists quickly, updating the max_length with the highest sum of lengths found so far.

## Method 3: Hashing With Prefix Sum

By combining a hash table with the prefix sum technique, we can significantly optimize the search for sublists matching the target sum. We store index information in the hash table to avoid iterating over all possible ending points for the second sublist.

Here’s an example:

def find_sublist_sums_with_hashing(lst, target_sum): max_length = 0 n = len(lst) prefix_sum = 0 sum_indices = {} # Create a hash table to store prefix sums and their indices for i, num in enumerate(lst): prefix_sum += num sum_indices.setdefault(prefix_sum, []).append(i) # Find non-overlapping sublists using the hash table current_sum = 0 for i, num in enumerate(lst): current_sum += num if current_sum == target_sum: max_length = max(max_length, i + 1) for index in sum_indices.get(current_sum + target_sum, []): if index > i: max_length = max(max_length, index - i) return max_length # Example list and target sum print(find_sublist_sums_with_hashing(example_list, target))

Output: 2

The code leverages a hash table to record the indices at which various prefix sums occur as we iterate over the list. For each element, it checks if the current sum equals the target, potentially updating the max_length. Additionally, it looks ahead for a second sublist starting after the current position that sums to the target and updates max_length if a longer non-overlapping pair is found.

## Method 4: Two-Pointers Technique

For a sorted list, the two-pointer technique can find pairs of nonoverlapping sublists by moving two pointers in the list based on the comparison of the sum of elements and the target sum. This method is efficient but requires the initial list to be sorted.

Here’s an example:

# Note: This method requires the list to be sorted def find_sublist_sums_two_pointers(lst, target_sum): # Sorting the list lst.sort() left = 0 right = len(lst) - 1 total_length = 0 while left < right: left_sum = lst[left] right_sum = lst[right] if left_sum + right_sum == target_sum: total_length += 2 left += 1 right -= 1 elif left_sum + right_sum < target_sum: left += 1 else: right -= 1 return total_length # Example list and target sum sorted_list = sorted(example_list) print(find_sublist_sums_two_pointers(sorted_list, target))

Output: 0

Assuming the provided list is sorted, the function `find_sublist_sums_two_pointers`

utilizes the two-pointer technique to efficiently find pairs of elements that add up to the target sum. The function increments the total_length by 2 whenever a matching pair is found and moves the pointers accordingly. This approach, however, is limited to pairs of elements rather than sublists.

## Bonus One-Liner Method 5: Using itertools and filter

Python’s `itertools`

library can be harnessed to generate combinations of sublist indices in a single line. We can then filter these for pairs that match the target sum. This one-liner is concise but not as readable or efficient as other methods.

Here’s an example:

from itertools import combinations # One-liner to find the max length of sublist sums find_max_length = lambda lst, target: max( (len(lst[a:b]) + len(lst[c:d]) for a, b in combinations(range(len(lst)+1), 2) for c, d in combinations(range(len(lst)+1), 2) if b <= c and sum(lst[a:b]) + sum(lst[c:d]) == target), default=0 ) # Example list and target sum print(find_max_length(example_list, target))

Output: 2

This one-liner uses a lambda function that iterates over all combinations of starting and ending indices for sublists using `itertools.combinations`

. It ensures that the sublists are nonoverlapping by checking that the second sublist starts after the first ends. The maximum length is computed by summing up the lengths of each valid pair of sublists.

## Summary/Discussion

**Method 1: Brute Force Approach.**Simple to implement. It can be inefficient for large lists with a time complexity of O(n^4).

**Method 2: Using Prefix Sum.**Optimizes the brute force by storing cumulative sums, improving time complexity. Still less efficient for larger lists.

**Method 3: Hashing With Prefix Sum.**Further optimizes by storing sums and their indices in a hash table; good for average-sized lists. Not the most space-efficient due to hash table usage.

**Method 4: Two-Pointers Technique.**Highly efficient for sorted lists with complexity O(n). Only works for pairs of elements, not larger sublists.

**Method 5: Using itertools and filter.**Elegant one-liner suited for small lists. Not practical for large datasets due to reduced readability and potential performance issues.