花花酱 LeetCode 2195. Append K Integers With Minimal Sum

You are given an integer array nums and an integer k. Append k unique positive integers that do not appear in nums to nums such that the resulting total sum is minimum.

Return the sum of the k integers appended to nums.

Example 1:

Input: nums = [1,4,25,10,25], k = 2
Output: 5
Explanation: The two unique positive integers that do not appear in nums which we append are 2 and 3.
The resulting sum of nums is 1 + 4 + 25 + 10 + 25 + 2 + 3 = 70, which is the minimum.
The sum of the two integers appended is 2 + 3 = 5, so we return 5.

Example 2:

Input: nums = [5,6], k = 6
Output: 25
Explanation: The six unique positive integers that do not appear in nums which we append are 1, 2, 3, 4, 7, and 8.
The resulting sum of nums is 5 + 6 + 1 + 2 + 3 + 4 + 7 + 8 = 36, which is the minimum. 
The sum of the six integers appended is 1 + 2 + 3 + 4 + 7 + 8 = 25, so we return 25.

Constraints:

• 1 <= nums.length <= 105
• 1 <= nums[i] <= 109
• 1 <= k <= 108

Solution: Greedy + Math, fill the gap

Sort all the numbers and remove duplicated ones, and fill the gap between two neighboring numbers.
e.g. [15, 3, 8, 8] => sorted = [3, 8, 15]
fill 0->3, 1,2, sum = ((0 + 1) + (3-1)) * (3-0-1) / 2 = 3
fill 3->8, 4, 5, 6, 7, sum = ((3 + 1) + (8-1)) * (8-3-1) / 2 = 22
fill 8->15, 9, 10, …, 14, …
fill 15->inf, 16, 17, …

Time complexity: O(nlogn)
Space complexity: O(1)