[Lowerbounds, Upperbounds]

On Geomblog, Piotr mentioned a problem that I think many of us can share: if we really have to pick just one to teach (as in his case), do we go with 2-4 trees or red-black trees?

For the most part, the pros and cons of the two approaches are quite, urr, balanced. But there has been one critical issue that tilt my preference towards RB. On the surface, the issue is “augmentation”, ala CLRS chapter 14 style. But it really is about the number of rotations used during insertions and deletions…

There is a fascinating history behind this, which is summarized by Ottmann and Wood up to 1990. It all started with a design from Olivie and a swift response from Tarjan. The rest of the juicy bits, well, I offer no spoiler. This post merely needs the following fact to make sense: in the worst case, some implementations of RB need only $O(1)$ rotations during restructuring and the remaining work are color changes; some implementations need logarithmic number of rotations.

But if we can afford a logarithmic number of color changes in the worst case, isn’t it just “a matter of constants” even if we have to do a logarithmic number of rotations as well?

In many cases, the answer is indeed yes, but there is no lack of interesting cases due to augmentations that make rotations “expensive”. For a starter: see section 13.4 of Bob’s textbook, pp 570-571 in the C++ 3rd edition, where this very issue is raised and the model answer is presented. Here is a favorite example of mine: Seidel and Aargon used this to highlight why treaps are preferable to splay trees in applications like segment trees. Let’s end with one more related to this post: McCreight’s priority search trees, used as the motivating example in Ottmann and Wood, also critically rely on search trees that use only $O(1)$ number of rotations to restructure. More precisely, doing $p$ rotations yields a bound of $O(p \log n)$. See, it’s not just in the constants!

I will finish this by saying that this doesn’t imply a landslide victory for RB. Conceptually, 2-4 trees are really hard to beat! In fact, even though I would be speaking RB in my mouth, I would be thinking about 2-4 at the back of my mind! (Kind of like some foreign language speakers, no? Ten years ago I was exactly like this, haha.) And really, if we ever get to more advanced stuff like supporting finger searches, I will pick 2-4 on most but not all days. Complications… complications…

Exercise: To understand how a 2-4 speaker would speak in RB, what is “black-height” in the 2-4 language?

Reference: Thomas Ottmann, Derick Wood: How to Update a Balanced Binary Tree with a Constant Number of Rotations. SWAT 1990: 122-131