Find median in O(1) in binary tree

Question

Suppose I have a balanced BST (binary search tree). Each tree node contains a special field count, which counts all descendants of that node + the node itself. They call this data structure order statistics binary tree.

This data structure supports two operations of O(logN):

rank(x) -- number of elements that are less than x
findByRank(k) -- find the node with rank == k

Now I would like to add a new operation median() to find the median. Can I assume this operation is O(1) if the tree is balanced?

I agree with Gir. But this is only true if the tree is completely balanced. — brano, Aug 10 '12 at 12:23

Ali Ferhat · Answer 1 · 2012-08-10T13:53:47.247

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Unless the tree is complete, the median might be a leaf node. So in general case the cost will be O(logN). I guess there is a data structure with requested properties and with a O(1) findMedian operation (Perhaps a skip list + a pointer to the median node; I'm not sure about findByRank and rank operations though) but a balanced BST is not one of them.

edited Aug 10 '12 at 13:53

answered Aug 10 '12 at 13:48

Ali Ferhat

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Yes, we can implement a _special_ data structure to find the median in O(1), e.g. 2 binary heaps or a skip list as you suggest. I wonder though if I can do it with the augmented balanced BST. – Michael Aug 10 '12 at 17:19

score 1 · Accepted Answer · answered Aug 10 '12 at 12:22

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If the tree is complete (i.e. all levels completely filled), yes you can.

answered Aug 10 '12 at 12:22

Andreas Brinck

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What if a tree is balanced but not `complete`? – Michael Aug 10 '12 at 12:25
you could use the count to decide which way to travel. not sure about O(1) in this case though – Gir Aug 10 '12 at 12:31
@Michael It depends on your definition of balanced, if you know both subtrees of the root has exactly the same number of children, the root will be the median. – Andreas Brinck Aug 10 '12 at 12:32
@AndreasBrinck "balanced" means that the heights of the left and right sub-trees of every node do not differ by more than 1. – Michael Aug 10 '12 at 12:38
@Michael In that case I don't think that finding the median will be an O(1) operation. – Andreas Brinck Aug 10 '12 at 12:43
It depends on what data you have available, if you know the number of nodes in each subtree it's O(log n), otherwise not. – Andreas Brinck Aug 10 '12 at 12:44

score 1 · Answer 3 · answered Aug 11 '12 at 20:39

In a balanced order statistics tree, finding the median is O(log N). If it is important to find the median in O(1) time, you can augment the data structure by maintaining a pointer to the median. The catch, of course, is that you would need to update this pointer during each Insert or Delete operation. Updating the pointer would take O(log N) time, but since those operations already take O(log N) time, the extra work of updating the median pointer does not change their big-O cost.

As a practical matter, this only makes sense if you do a lot of "find median" operations compared to the number of insertions/deletions.

If desired, you could reduce the cost of updating the median pointer during Insert/Delete to O(1) by using a (doubly) threaded binary tree, but Insert/Delete would still be O(log N).

Find median in O(1) in binary tree

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