How to find k nearest neighbors to the median of n distinct numbers in O(n) time?
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
Answer
No one seems to quite have this. Here's how to do it. First, find the median as described above. This is O(n). Now park the median at the end of the array, and subtract the median from every other element. Now find element k of the array (not including the last element), using the quick select algorithm again. This not only finds element k (in order), it also leaves the array so that the lowest k numbers are at the beginning of the array. These are the k closest to the median, once you add the median back in.