How to find k nearest neighbors to the median of n distinct numbers in O(n) time?
algorithm, median, nearest-neighbor, search, sorting
Solution
Actually, the answer is pretty simple. All we need to do is to select k elements with the smallest absolute differences from the median moving from m-1 to 0 and m+1 to n-1 when the median is at index m. We select the elements using the same idea we use in merging 2 sorted arrays.
Problem
I can use the median of medians selection algorithm to find the median in O(n). Also, I know that after the algorithm is done, all the elements to the left of the median are less that the median and all the elements to the right are greater than the median. But how do I find the k nearest neighbors to the median in O(n) time? If the median is n, the numbers to the left are less than n and the numbers to the right are greater than n. However, the array is not sorted in the left or the right sides. The numbers are any set of distinct numbers given by the user. The problem is from Introduction to Algorithms by Cormen, problem 9.3-7