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3695. Maximize Alternating Sum Using Swaps

Description

You are given an integer array nums.

You want to maximize the alternating sum of nums, which is defined as the value obtained by adding elements at even indices and subtracting elements at odd indices. That is, nums[0] - nums[1] + nums[2] - nums[3]...

You are also given a 2D integer array swaps where swaps[i] = [pi, qi]. For each pair [pi, qi] in swaps, you are allowed to swap the elements at indices pi and qi. These swaps can be performed any number of times and in any order.

Return the maximum possible alternating sum of nums.

 

Example 1:

Input: nums = [1,2,3], swaps = [[0,2],[1,2]]

Output: 4

Explanation:

The maximum alternating sum is achieved when nums is [2, 1, 3] or [3, 1, 2]. As an example, you can obtain nums = [2, 1, 3] as follows.

  • Swap nums[0] and nums[2]. nums is now [3, 2, 1].
  • Swap nums[1] and nums[2]. nums is now [3, 1, 2].
  • Swap nums[0] and nums[2]. nums is now [2, 1, 3].

Example 2:

Input: nums = [1,2,3], swaps = [[1,2]]

Output: 2

Explanation:

The maximum alternating sum is achieved by not performing any swaps.

Example 3:

Input: nums = [1,1000000000,1,1000000000,1,1000000000], swaps = []

Output: -2999999997

Explanation:

Since we cannot perform any swaps, the maximum alternating sum is achieved by not performing any swaps.

 

Constraints:

  • 2 <= nums.length <= 105
  • 1 <= nums[i] <= 109
  • 0 <= swaps.length <= 105
  • swaps[i] = [pi, qi]
  • 0 <= pi < qi <= nums.length - 1
  • [pi, qi] != [pj, qj]

Solutions

Solution 1

Thinking

The alternating sum is even indices minus odd indices. Allowed swaps make values interchangeable inside a connected component of indices.

Union-find builds those components. In a component with \(e\) even indices, assign the \(e\) largest values to even positions and the rest to odd ones.

Components are independent; the total is the maximum alternating sum. A sort (or a selection of the \(e\)-th largest) implements the assignment.

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