# 002616-Minimize-the-Maximum-Difference-of-Pairs

### Problem

https://leetcode.com/problems/minimize-the-maximum-difference-of-pairs/description/

You are given a 0-indexed integer array nums and an integer p. Find p pairs of indices of nums such that the maximum difference amongst all the pairs is minimized. Also, ensure no index appears more than once amongst the p pairs.

Note that for a pair of elements at the index i and j, the difference of this pair is |nums\[i] - nums\[j]|, where |x| represents the absolute value of x.

Return the minimum maximum difference among all p pairs. We define the maximum of an empty set to be zero.

Example 1:

Input: nums = \[10,1,2,7,1,3], p = 2 Output: 1 Explanation: The first pair is formed from the indices 1 and 4, and the second pair is formed from the indices 2 and 5. The maximum difference is max(|nums\[1] - nums\[4]|, |nums\[2] - nums\[5]|) = max(0, 1) = 1. Therefore, we return 1.

Example 2:

Input: nums = \[4,2,1,2], p = 1 Output: 0 Explanation: Let the indices 1 and 3 form a pair. The difference of that pair is |2 - 2| = 0, which is the minimum we can attain.

Constraints:

1 <= nums.length <= 105 0 <= nums\[i] <= 109 0 <= p <= (nums.length)/2

### Solution

binary search,

1. sort array
2. check each mid number can form p pairs
3. find the min value and return

```python
class Solution:
    def minimizeMax(self, nums: List[int], p: int) -> int:
        n = len(nums)
        def can_form_pairs(target, numPairs):
            idx, count = 0, 0
            while idx < n-1:
                if nums[idx+1] - nums[idx] <= target:
                    count += 1
                    idx += 2
                else:
                    idx += 1
            return count >= numPairs
        
        nums.sort()
        left, right = 0, nums[-1]-nums[0]
        while left <= right:
            mid = (left+right)//2
            if can_form_pairs(mid, p):
                right = mid - 1
            else:
                left = mid + 1
        return left

```
