Some SoCG'10 papers

SoCG is over for this year. In general I thought that the quality of the accepted papers was very high, and I really enjoyed many of the presentations and the results. There has already been numerous reports from SoCG (Sorelle I, II and III, David, Sariel and Suresh), and below I'm just going to briefly mention three of the ones that I found the most interesting - in order of presentation.

Timothy Chan - Optimal Partition Trees (Paper).
Timothy improves on a result from 1992 by Matousek. He presents a simpler construction with a better preprocessing time. There are already numerous blog comments on it (Sariel and Sorelle).

Joe Mitchell - A constant factor approximation algorithm for TSP with neighbourhoods in the plane (paper).
This is a problem I've been working on back and forth for a long time. Given a collection of objects (usually polygons) in the plane, called neighbourhoods, find the shortest tour that visits all neighbourhoods. In 2002 we presented a constant factor approximation algorithm for fat disjoint objects of arbitrary size. This was the first constant factor approximation algorithm for arbitrary size objects. Joe has several results on the topic and in his SoCG paper he presents the first constant-factor approximation algorithm for TSPN on an arbitrary set of disjoint, connected neighbourhoods in the plane. This has been an open problem since the mid-90's and Joe's paper is a big step forward. Of course, ultimately we would like a constant-factor approximation for the general case, not necessarily disjoint neighbourhoods. He has several interesting general observations in his paper. One is that skinny neighbourhoods are not longer a big problem. Using “directional hulls”, which are fat, for an appropriate choice of one of two coordinate systems. Then one can partition the regions into two sets (“blue” regions and “red” regions). Separately find approximating tours of each set, and then use the Combination Lemma (Arkin and Hassin) to combine into a tour of the entire set.

Anne Driemel, Sariel Har-Peled and Carola Wenk - Approximating the Frechet distance for realistic curves in near linear time (paper).
This is probably my favourite paper at SoCG this year. It's an important problem (imho) and it's great to finally see a subquadratic time algorithm with reasonable assumptions. The aim is to calculate the Frechet distance between two curves. They introduce a notion of k-bounded curvature which means that in a ball of radius r the length of the trajectory inside that ball is bounded by k*r (this family of curves is closed under simplification, a property that is very useful in practice). This sounds like a reasonable assumption for many applications (although not for trajectories generated from football players). They present a (1+eps)-approximation algorithm to compute the Frechet distance between two curves with k-bounded curvature and prove that this can be done in near-linear time. The key idea is that after the curves have been simplified the complexity of the reachable free space is linear for fixed k and epsilon (the free space diagram is the standard tool for computing the Frechet distance between curves but has quadratic complexity).