SODA 2007: Day 0

Getting into New Orleans at 1 am because of "mechanical trouble" meant that I haven't been at my best so far. But I've already heard one amazing talk today.

Luc Devroye gave the ANALCO plenary lecture on "Weighted Heights of Random Trees", based on work with his students Erin McLeish and Nicolas Broutin. After having sat through many talks with titles like this, I generally approach them with great caution and with a clear escape route. But...

This was an amazing exposition of a topic that could have become dry and terse, and essentially incomprehensible, within a slide or two. He had jokes, (that were funny), a global plan for the material, enough technical material that I went away feeling like I'd learnt something, and intuition galore. And the work itself is very beautiful.

So what was it all about ? The problem is really quite simple to state. Suppose I give you a (random) weighted binary tree, where nodes attach to parents randomly, and edges may have weights chosen randomly. What is the maximum height of such a tree ?

The standard application for such a tool is in analyzing binary search trees. The height of a such a tree controls the running time of an algorithm that needs to use it. And there's now a vast literature analyzing both the asymptotics of the height distribution (basically it's sharply concentrated around 2 log n) and the specific constants (the maximum height of a random binary search tree is roughly 4.3 log n, and the minimum is around 0.37 log n).

The "master goal" that Devroye described in his talk was this: Suppose I have a general way of attaching nodes to parents (that leads to a general distribution on subtree sizes), and a general way of attaching weights to edges (rather than being deterministically 1 for binary search trees). Such a general model captures the analysis of tries (trees on strings that are very important in text searching), geometric search structures like k-d trees, and even restricted preferential attachment models in social network analysis (Think of the edges as hyperlinks, and the height of the tree as the diameter of a web tree).

Is there a generic theorem that can be applied to all of these different situations, so that you can plug in a set of distributions that describes your process, and out pops a bound on the height of your tree ? It turns out that you can (with some technical conditions). The method uses two-dimensional large-deviation theory: can you estimate the probability of a sum of random variables being bounded by some function, while at the same time ensuring that some other sum of random variables (that might depend slightly on the first) is also bounded ?

An example of a 1D large deviation result is of course a Chernoff bound. Devroye showed that a 2D large deviation bound for the height of such trees can be expressed in a similar form using the so-called Cramér exponent, something that will probably not be surprising to experts in large deviation theory. After that, the analysis for any tree process becomes a whole lot easier. You have to analyze the corresponding Cramér function for your distributions, and a bound (with a constant; no big-O nonsense here!) pops out.

He also talked about a neat extension of this method to analyzing the "skinnyness" of k-d tree decompositions, showing that for a kind of "relaxed" k-d tree construction, the skinniest cell can be extremely skinny (having a super-linear aspect ratio). It's the kind of result that I imagine would be very difficult to prove without such a useful general theorem.