Minimum Difference in Sums After Removal of Elements

arraydynamic programmingheap (priority queue)

Problem Statement

You are given a 0-indexed integer array nums consisting of 3 * n elements.

You are allowed to remove any subsequence of elements of size exactly n from nums . The remaining 2 * n elements will be divided into two equal parts:

The first n elements belonging to the first part and their sum is sum _first .

The next n elements belonging to the second part and their sum is sum _second .

The difference in sums of the two parts is denoted as sum _first - sum _second .

For example, if sum _first = 3 and sum _second = 2 , their difference is 1 .

Similarly, if sum _first = 2 and sum _second = 3 , their difference is -1 .

Return the minimum difference possible between the sums of the two parts after the removal of n elements .

Examples

Example 1

Input: nums = [3,1,2]
Output: -1
Here, nums has 3 elements, so n = 1. 
Thus we have to remove 1 element from nums and divide the array into two equal parts.
- If we remove nums[0] = 3, the array will be [1,2]. The difference in sums of the two parts will be 1 - 2 = -1.
- If we remove nums[1] = 1, the array will be [3,2]. The difference in sums of the two parts will be 3 - 2 = 1.
- If we remove nums[2] = 2, the array will be [3,1]. The difference in sums of the two parts will be 3 - 1 = 2.
The minimum difference between sums of the two parts is min(-1,1,2) = -1.

Example 2

Input: nums = [7,9,5,8,1,3]
Output: 1
Here n = 2. So we must remove 2 elements and divide the remaining array into two parts containing two elements each.
If we remove nums[2] = 5 and nums[3] = 8, the resultant array will be [7,9,1,3]. The difference in sums will be (7+9) - (1+3) = 12.
To obtain the minimum difference, we should remove nums[1] = 9 and nums[4] = 1. The resultant array becomes [7,5,8,3]. The difference in sums of the two parts is (7+5) - (8+3) = 1.
It can be shown that it is not possible to obtain a difference smaller than 1.

Constraints

nums.length == 3 * n

1 <= n <= 10 ^5

1 <= nums[i] <= 10 ^5

😤

Hard

Difficulty

Topic Info

ModuleDSA

CategoryArray

Sub-topicDynamic Programming