**π‘ Problem Formulation:** The task is to calculate the expected sum of all possible subarrays of a given array. For instance, given the array `[1, 2, 3]`

, the subarrays are `[1], [2], [3], [1, 2], [2, 3],`

and `[1, 2, 3]`

. The expected sum is the total of the sums of these subarrays. The objective is to find this sum efficiently using various methods in Python.

## Method 1: Brute Force Approach

This method involves generating all possible subarrays of the given array and then calculating the sum of elements in each subarray. Although straightforward, this approach is not efficient for large arrays as it has a time complexity of O(n^2).

Here’s an example:

def sum_of_subarrays(arr): total_sum = 0 n = len(arr) for i in range(n): for j in range(i, n): total_sum += sum(arr[i:j+1]) return total_sum arr = [1, 2, 3] print(sum_of_subarrays(arr))

Output: 20

This piece of code systematically iterates through the array, generating all possible subarrays. It then calculates the sum of each of these subarrays and adds it to the total sum, which is returned at the end.

## Method 2: Using Cumulative Sum

By first calculating the cumulative sum of the original array, this method reduces the problem. It allows us to quickly determine the sum of any subarray using the precomputed sums. This method is more efficient than the brute force with a time complexity of O(n).

Here’s an example:

def sum_of_subarrays(arr): n = len(arr) cumulative_sum = [0] * (n + 1) for i in range(1, n + 1): cumulative_sum[i] = cumulative_sum[i - 1] + arr[i - 1] total_sum = 0 for i in range(n): for j in range(i, n): total_sum += cumulative_sum[j + 1] - cumulative_sum[i] return total_sum arr = [1, 2, 3] print(sum_of_subarrays(arr))

Output: 20

In this snippet, we create a cumulative sum list that allows us to compute subarray sums quickly. The total sum of all subarrays is then calculated using this cumulative sum.

## Method 3: Efficient Mathematical Approach

This efficient method capitalizes on the observation that each element in the array contributes to several subarrays. By using a formula that takes into account the position of each element, this approach calculates the total expected sum of subarrays in O(n) time.

Here’s an example:

def sum_of_subarrays(arr): total_sum = 0 n = len(arr) for i, val in enumerate(arr): # Element 'val' appears in (i+1) * (n-i) subarrays. total_sum += val * (i+1) * (n-i) return total_sum arr = [1, 2, 3] print(sum_of_subarrays(arr))

Output: 20

This code leverages the fact that an element at index `i`

is included in `(i+1)*(n-i)`

subarrays. Multiplied by the element value, this gives us the total contribution of that element to the sum of all subarrays.

## Method 4: Using itertools Combinations

This Pythonic way utilizes the `itertools.combinations`

library to generate all possible subarrays without explicit nested loops. We then sum each subarray and aggregate these sums. This method offers concise code but doesn’t improve the time complexity significantly.

Here’s an example:

from itertools import combinations def sum_of_subarrays(arr): return sum(sum(arr[i:j+1]) for i, j in combinations(range(len(arr) + 1), 2)) arr = [1, 2, 3] print(sum_of_subarrays(arr))

Output: 20

The combination function is used to generate the start and end indices (`i, j`

) for all subarrays. The sum of each subarray is computed using list slicing and aggregated to get the total sum.

## Bonus One-Liner Method 5: Lambda and reduce

For a touch of functional programming flair, this one-liner uses the `reduce`

function in combination with a lambda expression to calculate the total sum of subarrays. It’s a neat trick but not necessarily recommended for readability or efficiency.

Here’s an example:

from functools import reduce arr = [1, 2, 3] total_sum = reduce(lambda acc, x: acc + x * (i+1) * (n-i), arr, 0) print(total_sum)

Output: 20

The reduce function is used to apply a lambda that computes the contribution of each array element to the sum of subarrays, with the running total being passed as the accumulator.

## Summary/Discussion

**Method 1:**Brute Force Approach. Straightforward and easy to implement, but inefficient for large arrays due to its O(n^2) time complexity.**Method 2:**Using Cumulative Sum. More efficient than brute force, offers O(n) complexity using precomputed sums, still not optimal for very large arrays.**Method 3:**Efficient Mathematical Approach. Highly efficient with O(n) complexity, leverages mathematical insights for quick computation.**Method 4:**Using itertools Combinations. Pythonic and concise but does not offer a significant improvement in time complexity over the brute force approach.**Bonus Method 5:**Lambda and reduce. A functional programming one-liner that is neat but may not be as readable or efficient as other methods.