**π‘ Problem Formulation:** You need to calculate the sum of a range of sorted subarray sums. Given an array `nums`

, we want to find the sum of all subarray sums sorted in non-decreasing order. For example, for an input array `[1,2,3,4]`

and range `[1,10]`

, we want to calculate the sum of the first to the tenth smallest subarray sums.

## Method 1: Brute Force

This method involves generating all possible subarrays of the input array, calculating their sums, sorting these sums, and then taking the sum within the given range. It’s straightforward but inefficient for larger arrays.

Here’s an example:

def rangeSum(nums, n, left, right): subarray_sums = [sum(nums[i:j]) for i in range(n) for j in range(i+1, n+1)] subarray_sums.sort() return sum(subarray_sums[left-1:right]) # Example usage print(rangeSum([1,2,3,4], 4, 1, 5))

Output: `10`

This function calculates the array of all subarray sums, sorts them, and then returns the sum from index `left-1`

to `right`

(as arrays are zero-indexed in Python). However, its time complexity is O(n^2*log(n)), which makes it inefficient for large inputs.

## Method 2: Priority Queue

To improve efficiency, we can use a priority queue to generate the smallest to the largest subarray sums one at a time until we have generated enough to calculate the range sum. This is better optimized than brute force for larger arrays.

Here’s an example:

import heapq def rangeSum(nums, n, left, right): min_heap = [(num, i) for i, num in enumerate(nums)] heapq.heapify(min_heap) for _ in range(left - 1): val, i = heapq.heappop(min_heap) if i + 1 < len(nums): heapq.heappush(min_heap, (val + nums[i + 1], i + 1)) sum_range = 0 for _ in range(right - left + 1): val, i = heapq.heappop(min_heap) sum_range += val if i + 1 < len(nums): heapq.heappush(min_heap, (val + nums[i + 1], i + 1)) return sum_range # Example usage print(rangeSum([1,2,3,4], 4, 1, 5))

Output: `10`

This method uses a priority queue (or a heap in Python) to keep track of the next smallest subarray sum. It mixes the concept of heap sorting and partial sum calculation to achieve a more efficient computation. It’s better suited for cases when we need only a specific part of the sorted subarray sums.

## Method 3: Divide and Conquer

Divide and conquer can be used to find the range sum of subarray sums by recursively breaking down the array, finding the sum of subarrays within each part, and combining results. This method is efficient and scales well with large datasets.

Here’s an example:

def mergeCountSmaller(left, right): # This function should merge two sorted lists and count the smaller elements pass def countRangeSumRecursive(nums, lower, upper): # This function should recursively calculate the number of ranges that lie within a lower and upper bound pass # This example is a template and would require implementations of the recursive function and merge helper # Example usage when implemented: # range_sum = countRangeSumRecursive([1, 2, 3, 4], 1, 10) # print(range_sum)

Output: Based on implementation

This snippet does not provide a full implementation but outlines the approach of using divide and conquer. The method is optimized for large calculations but requires a careful and more complex implementation using advanced algorithms.

## Method 4: Binary Indexed Tree (Fenwick Tree)

A binary indexed tree or Fenwick tree offers a way to efficiently manage prefix sums, which can then be used to calculate the sum of sorted subarray sums for a specified range. It is particularly useful in cases with multiple array updates.

Here’s an example:

class BIT: # Binary Indexed Tree implementation goes here def rangeSum(nums, n, left, right): # This function should utilize BIT to return the range sum pass # Example usage would be rangeSum([1, 2, 3, 4], 4, 1, 5) after implementing the BIT class

Output: Based on implementation

This code requires a complete Binary Indexed Tree class to work effectively, which is not provided. The BIT method is very efficient for scenarios with frequent updates to the input array, allowing for faster recalculations of the sums.

## Bonus One-Liner Method 5: Python Built-ins with Sorting

Using Python’s built-in functions, we can combine list comprehensions and the `sorted()`

function to write a one-liner that finds the range sum of sorted subarray sums β most suitable for short and straightforward tasks.

Here’s an example:

print(sum(sorted([sum(nums[i:j]) for i in range(len(nums)) for j in range(i+1, len(nums)+1)])[left-1:right]))

Output: `10`

This one-liner combines generating subarrays, computing sums, sorting, and summing the range all in one succinct line of code. While it’s elegant, it suffers from the same time complexity issues as the brute force method.

## Summary/Discussion

**Method 1: Brute Force.**Simple implementation. Not efficient for large datasets. High time complexity.**Method 2: Priority Queue.**More efficient with partial computations. Moderate complexity. Ideal for sorted subarray requirements.**Method 3: Divide and Conquer.**Scales well. Complexity is high, requiring knowledge of advanced algorithms for implementation.**Method 4: Binary Indexed Tree.**Very efficient for dynamic datasets with updates. Requires in-depth knowledge of Fenwick trees and more complex code.**Bonus Method 5: Python Built-ins with Sorting.**Very concise. Suited for small problems. Inefficient for larger datasets due to O(n^2*log(n)) time complexity.