Regular triangulations of dynamic sets of points

The Delaunay triangulations of a set of points are a class of triangulations which play an important role in a variety of different disciplines of science. Regular triangulations are a generalization of Delaunay triangulations that maintain both their relationship with convex hulls and with Voronoi diagrams. In regular triangulations, a real value, its weight, is assigned to each point. In this paper a simple data structure is presented that allows regular triangulations of sets of points to be dynamically updated, that is, new points can be incrementally inserted in the set and old points can be deleted from it. The algorithms we propose for insertion and deletion are based on a geometrical interpretation of the history data structure in one more dimension and use lifted flips as the unique topological operation. This results in rather simple and efficient algorithms. The algorithms have been implemented and experimental results are given.Postprint (published version

Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions

We describe the implementation of algorithms to construct and maintain three-dimensional dynamic Delaunay triangulations with kinetic vertices using a three-simplex data structure. The code is capable of constructing the geometric dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points is triangulated. Time evolution of the triangulation is not only governed by kinetic vertices but also by a changing number of vertices. We use three-dimensional simplex flip algorithms, a stochastic visibility walk algorithm for point location and in addition, we propose a new simple method of deleting vertices from an existing three-dimensional Delaunay triangulation while maintaining the Delaunay property. The dual Dirichlet tessellation can be used to solve differential equations on an irregular grid, to define partitions in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly nomenclature), referee suggestions included, typos corrected, bibliography update

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

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

The polytope of non-crossing graphs on a planar point set

For any finite set \A of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set \A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on \A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension $2n_i +n -3$ where $n_i$ is the number of points of \A in the interior of \conv(\A). The vertices of this polytope are all the pseudo-triangulations of \A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has been reshape