Sunday, January 06, 2013

SODA 2013, Part I/n

On twitter, it's common to post longer thoughts in 140 character bursts. If you don't know how long the thought will be, you label them as 1/n, 2/n, and so on, so as not to exceed an arbitrary limit, but also imply that there is a finite limit. 

So we're back in the Big Easy for SODA. This time the conference has merged ALENEX and ANALCO with the main program, so at least for today and tomorrow, we have four parallel tracks. Having just come back from NIPS, SODA feels like a small cozy cocktail party in comparison (ed: not that I know anything about cocktail parties: I have small children)

The highlight was Bob Sedgewick's tribute to Philippe Flajolet. I've talked about their work on analytic combinatorics before, but it was nice to hear some of it again. Flajolet's contributions to the field are immense and valuable. He did big data before it was cool: his $\ell_0$ estimation work with Nigel Martin is now a seminal paper in the world of streaming, and his HyperLogLog paper on counting a stream (with Fusy, Gandouet and Meunier) is the best-in-class on this problem. Bob quoted one of Flajolet's students as saying, "Read any of his papers. You will learn something".

I chaired a geometry session in the morning, and one of the papers there caught my eye a while back. The Fréchet distance problem (commonly known as the dog-walking problem) is the problem of computing a monotone mapping between two curves that has minimim maximum length (you can think of this as a man on one curve walking a dog on another, and asking for the minimum leash length when both man and dog can walk forward at arbitrary speeds).

There's a relatively straightforward dynamic program that can compute the Fréchet distance between two polygonal chains in time $O(nm\log (nm))$ (if $n$ and $m$ are the number of links in the two chains). But it's been a big open problem to figure out whether this can be done in subquadratic time. The paper, (by Agarwal, Ben Avraham, Kaplan and Sharir) shows that for the discrete version of the problem (that Rinat Ben Avraham in her talk calls the frog hopping problem: the frogs hop from vertex to vertex) you can get a slightly subquadratic time algorithm. The main idea involves adapting "string-like" methods for the problem, which is hard because the "alphabet" is infinitely large.

It will be interesting to see if this spurs more progress. There's already a newer paper by Buchin et al that gets a slightly improved (but still super-quadratic) algorithm for the continuous Fréchet distance (i.e the original problem). If for no other reason, please click the link to see their awesome title.

Martin Grohe gave an award talk for his work with Kawarabayashi and Reed on an improved algorithm for graph minor decompositions. The problem is as follows: given a graph G and a graph H supposedly excluded by G, compute a decomposition of G promised by the graph minor theorem, or produce an H-minor.

The improvement reduces the dependency on the size of $G$ to quadratic (from cubic) and makes use of the wonderful and mysterious Courcelle's theorem. The dependence on the size of $H$ is equally mysterious (at least Martin declined to tell us), but is nowhere near as wonderful.

Please post papers you thought were interesting (and why) in the comments, and I'll add them as updates.

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