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

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.

The median-of-medians probably doesn't help much in finding the nearest neighbours, at least for large n. True, you have each column of 5 partitioned around it's median, but this isn't enough ordering information to solve the problem.

I'd just treat the median as an intermediate result, and treat the nearest neighbours as a priority queue problem...

Once you have the median from the median-of-medians, keep a note of it's value.

Run the heapify algorithm on all your data - see Wikipedia - Binary Heap. In comparisons, base the result on the difference relative to that saved median value. The highest priority items are those with the lowest ABS(value - median). This takes O(n).

The first item in the array is now the median (or a duplicate of it), and the array has heap structure. Use the heap extract algorithm to pull out as many nearest-neighbours as you need. This is O(k log n) for k nearest neighbours.

So long as k is a constant, you get O(n) median of medians, O(n) heapify and O(log n) extracting, giving O(n) overall.

med=Select(A,1,n,n/2)   //finds the median

for i=1 to n

q=Select(B,1,n,k) //get the kth smallest difference

for i=1 to n
   if B[i]<=q 
     C[j]=A[i] //A[i], the real value should be assigned instead of B[i] which is only the difference between A[i] and median.
return C