Maximum difference between two subsets of m elements

Given an array of n integers and a number m, find the maximum possible difference between two sets of m elements chosen from given array.

Examples: 

Input : arr[] = 1 2 3 4 5
m = 4
Output : 4
The maximum four elements are 2, 3, 4 and 5. The minimum four elements are 1, 2, 3 and 4. The difference between two sums is (2 + 3 + 4 + 5) - (1 + 2 + 3 + 4) = 4

Input : arr[] = 5 8 11 40 15
m = 2
Output : 42
The difference is (40 + 15) - (5 + 8).

[Naive Solution] - Using Sorting - O(n Log n) Time and O(1) Space

The idea is to first sort the array, then find sum of first m elements and sum of last m elements. Finally return difference between two sums.

4

[Naive Solution] - Using Heap - O(n Log m) Time and O(m) Space

The idea is based on the solution discussed in k largest elements in an array

• Create a Min Heap to Store m largest. We first push first m elements. For remaining n-m elements, we compare every element with the heap top. If Heap top is smaller, we insert the new element in the min heap
• In the same way, create a Max Heap to Store m smallest.
• Find the sums of elements in both the heaps and return the difference of two sums.

4

Time Complexity: O(n * log m), this solution can work in O(m + (n-m) Log m) as build heap take linear time.