Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n \log^3 n)$ space and polynomial preprocessing time, and answers queries in $O(\log n)$ time. These are the first solutions to the problem with $O(\log n)$ query time and $o(n^2)$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $O(\log n)$ amortized pointer changes, in addition to $O(\log n)$-time point-location queries, even though every such update may make $\Theta(n)$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in $o(n)$ amortized pointer changes per operation while keeping $O(\log n)$-time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in Algorithmic

A minimalistic approach for fast computation of geodesic distances on triangular meshes

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes.Comment: Preprint submitted to Computers & Graphic

Proximity problems on line segments spanned by points

AbstractFinding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S∖{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space

Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.Comment: Full version of L. Rademacher, S. Vempala: Dispersion of Mass and the Complexity of Randomized Geometric Algorithms. Proc. 47th IEEE Annual Symp. on Found. of Comp. Sci. (2006). A version of it to appear in Advances in Mathematic

Moving Walkways, Escalators, and Elevators

We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful. We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set.Comment: 16 pages. Presented at XII Encuentros de Geometria Computacional, Valladolid, Spai