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

On the Complexity of Randomly Weighted Voronoi Diagrams

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal etal [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems

The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of some parts; correction of several inaccuracies; improvement of some proofs and figures; added references; modification of the title; the paper is long but more than half of it is composed of proofs and references: it is sufficient to look at pages 5, 7--11 in order to understand the algorith

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

Optimal relay location and power allocation for low SNR broadcast relay channels

We consider the broadcast relay channel (BRC), where a single source transmits to multiple destinations with the help of a relay, in the limit of a large bandwidth. We address the problem of optimal relay positioning and power allocations at source and relay, to maximize the multicast rate from source to all destinations. To solve such a network planning problem, we develop a three-faceted approach based on an underlying information theoretic model, computational geometric aspects, and network optimization tools. Firstly, assuming superposition coding and frequency division between the source and the relay, the information theoretic framework yields a hypergraph model of the wideband BRC, which captures the dependency of achievable rate-tuples on the network topology. As the relay position varies, so does the set of hyperarcs constituting the hypergraph, rendering the combinatorial nature of optimization problem. We show that the convex hull C of all nodes in the 2-D plane can be divided into disjoint regions corresponding to distinct hyperarcs sets. These sets are obtained by superimposing all k-th order Voronoi tessellation of C. We propose an easy and efficient algorithm to compute all hyperarc sets, and prove they are polynomially bounded. Using the switched hypergraph approach, we model the original problem as a continuous yet non-convex network optimization program. Ultimately, availing on the techniques of geometric programming and $p$-norm surrogate approximation, we derive a good convex approximation. We provide a detailed characterization of the problem for collinearly located destinations, and then give a generalization for arbitrarily located destinations. Finally, we show strong gains for the optimal relay positioning compared to seemingly interesting positions.Comment: In Proceedings of INFOCOM 201

On the hausdorff and other cluster Voronoi diagrams

The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand various generalizations of this simple concept. While many generalized Voronoi diagrams have been well studied, many others still have unsettled questions. An example of the latter are cluster Voronoi diagrams, whose sites are sets (clusters) of objects rather than individual objects. In this dissertation we study certain cluster Voronoi diagrams from the perspective of their construction algorithms and algorithmic applications. Our main focus is the Hausdorff Voronoi diagram; we also study the farthest-segment Voronoi diagram, as well as certain special cases of the farthest-color Voronoi diagram. We establish a connection between cluster Voronoi diagrams and the stabbing circle problem for segments in the plane. Our results are as follows. (1) We investigate the randomized incremental construction of the Hausdorff Voronoi diagram. We consider separately the case of non-crossing clusters, when the combinatorial complexity of the diagram is O(n) where n is the total number of points in all clusters. For this case, we present two construction algorithms that require O(n log2 n) expected time. For the general case of arbitrary clusters, we present an algorithm that requires O((m + n log n) log n) expected time and O(m + n log n) expected space, where m is a parameter reflecting the number of crossings between clusters' convex hulls. (2) We present an O(n) time algorithm to construct the farthest-segment Voronoi diagram of n segments, after the sequence of its faces at infinity is known. This augments the well-known linear-time framework for Voronoi diagram of points in convex position, with the ability to handle disconnected Voronoi regions. (3) We establish a connection between the cluster Voronoi diagrams (the Hausdorff and the farthest-color Voronoi diagram) and the stabbing circle problem. This implies a new method to solve the latter problem. Our method results in a near-optimal O(n log2 n) time algorithm for a set of n parallel segments, and in an optimal O(n log n) time algorithm for a set of n segments satisfying some other special conditions. (4) We study the farthest-color Voronoi diagram in special cases considered by the stabbing circle problem. We prove O(n) bound for its combinatorial complexity and present an O(nlogn) time algorithm to construct it

Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures