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Suppose I have an AVL tree with a pointer to the minimal element. I'd like to conduct a search for some key x, which is the $k$-smallest key in the entire tree. I can do this by "climbing" up the tree's left branch, comparing x with the current key. As long as

(x > curr.key) && (x > curr.parent.key)

I keep climbing up, but once the second condition is violated, I slide down to the current right-subtree, and from there on it's just a standard BST search.

The claim is that the worst case complexity is always $O(\log k)$, for any $k$. But I can't convince myself this is accurate: if x is larger than the tree's root's key (the median, or equivalently $k > {n\over 2}$), that implies I must have traversed the entire left branch, which for a balanced tree is $O(\log n)$ - and only then I can find x in depth of $O(\log k)$.

Am I looking at this the wrong way?

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  • $\begingroup$ "The claim is that ...". Can you add a url or reference to the place of that claim? $\endgroup$ – Apass.Jack Dec 6 '18 at 19:44
  • $\begingroup$ It looks like you have understood the whole situation correctly except a minor glitch somewhere. By the way, if $n\ge k>n/2$, then $O(\log n)$ is $O(\log k)$ $\endgroup$ – Apass.Jack Dec 6 '18 at 19:47
  • $\begingroup$ @Apass.Jack The claim was from my class... and no further information on that topic is provided, either. I understand your note, but still - climbing up the tree isn't done in a constant time, the height from the minimum is still $O (\log n)$. The technicality for bigger k's doesn't hold, it seems. So you're saying there's no way to achieve $O(\log n)$ without comparing with the root first? $\endgroup$ – gbi1977 Dec 6 '18 at 20:57
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Yes, you might be looking it the wrong way.

If $x$ is larger than the tree's root's key (the median, or equivalently $k>n/2$), that implies I must have traversed the entire left branch, which for a balanced tree is $O(\log n)$ - and only then I can find $x$ in depth of $O(\log k)$.

First, let us be accurate. The root's key may not be the median, although it sits roughly in the middle.

Let $r$ be the root node. Let $n$ be the number of nodes in the whole tree and $n_l$ be the number of nodes in the left subtree of $r$. Let $h_l$ be the height of that subtree and $h$ be the height of the tree, so $h\le h_l+2$ since an AVL tree is height balanced.

Suppose $x$ is larger than the root's key. That means $$n_l < k \le n.$$ We also have, according to the properties of an AVL tree. $$\log_2 (n+1)\le h \le h_l+2 \le c\log_2(n_l+2)+b +2\lt c\log_2(k+2)+b +2$$ where $c\approx 1.44$, $b\approx -0.328$. So, for all $k\ge3$, $$\log_2(n)\le2\log_2(k)$$

So if you have traversed the entire (height) of the left branch in order find the $k$-th smallest element, you will use $O(\log n) = O(\log k)$ steps.

In fact, if we only look at the subtree whose root is the highest node that we will climb to in search of the $k$-th smallest element, we have just proved that the worst case complexity is always $O(\log k)$

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