Problem Description
The distance of a pair of integers a and b is defined as the absolute difference between a and b. Given an integer array nums and an integer k, return the k-th smallest distance among all the pairs nums[i] and nums[j] where 0 <= i < j < nums.length.
Key Insights
- We need to calculate the distance for all pairs in the given array.
- Distances can range from 0 (when two numbers are the same) to the maximum difference between the smallest and largest numbers in the array.
- A binary search can be used to find the k-th smallest distance efficiently.
- The number of pairs that have a distance less than or equal to a certain value can be counted using a two-pointer technique on a sorted version of the array.
Space and Time Complexity
Time Complexity: O(n log(max_distance) + n log(n))
Space Complexity: O(n)
Solution
We can approach the problem using binary search to find the k-th smallest distance. First, we sort the array to make it easier to calculate distances. Then, we define our search space from 0 to the maximum possible distance (the difference between the maximum and minimum numbers in the array). For each potential distance in our search space, we count how many pairs have a distance less than or equal to this value using a two-pointer technique. We adjust our search space based on whether we have found enough pairs.