Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm

A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs

We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g, where f(n) is the best approximation factor achievable in polynomial time on arbitrary n-vertex graphs. In particular, the O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation algorithm for genus-g graphs. Our result improves the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g; ours is the first approximation algorithm for graphs with bounded non-orientable genus. Moreover, using recent progress on approximating the genus of a graph, our O(log(g) / loglog(g))-approximation can be implemented even without an embedding when the input graph has bounded degree. In contrast, the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a genus-g embedding as part of the input. Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on graphs of genus g, with running time 2^O(g)*n^O(1)

Flipping Cubical Meshes

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange