How to find k nearest neighbors to the median of n distinct numbers in O(n) time?

Question

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 les

Answer 1

You could use a non-comparison sort, such as a radix sort, on the list of numbers L, then find the k closest neighbors by considering windows of k elements and examining the window endpoints. Another way of stating "find the window" is find i that minimizes abs(L[(n-k)/2+i] - L[n/2]) + abs(L[(n+k)/2+i] - L[n/2]) (if k is odd) or abs(L[(n-k)/2+i] - L[n/2]) + abs(L[(n+k)/2+i+1] - L[n/2]) (if k is even). Combining the cases, abs(L[(n-k)/2+i] - L[n/2]) + abs(L[(n+k)/2+i+!(k&1)] - L[n/2]). A simple, O(k) way of finding the minimum is to start with i=0, then slide to the left or right, but you should be able to find the minimum in O(log(k)).

The expression you minimize comes from transforming L into another list, M, by taking the difference of each element from the median.

m=L[n/2]
M=abs(L-m)

i minimizes M[n/2-k/2+i] + M[n/2+k/2+i].