1,231 research outputs found

    Fitting Voronoi Diagrams to Planar Tesselations

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    Given a tesselation of the plane, defined by a planar straight-line graph GG, we want to find a minimal set SS of points in the plane, such that the Voronoi diagram associated with SS "fits" \ GG. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of GG, assuming that the smallest angle in GG is constant.Comment: 14 pages, 8 figures, 1 table. Presented at IWOCA 2013 (Int. Workshop on Combinatorial Algorithms), Rouen, France, July 201

    Effects of Boundary Conditions on Single-File Pedestrian Flow

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    In this paper we investigate effects of boundary conditions on one dimensional pedestrian flow which involves purely longitudinal interactions. Qualitatively, stop-and-go waves are observed under closed boundary condition and dissolve when the boundary is open. To get more detailed information the fundamental diagrams of the open and closed systems are compared using Voronoi-based measurement method. Higher maximal specific flow is observed from the pedestrian movement at open boundary condition

    On the Complexity of Randomly Weighted Voronoi Diagrams

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    In this paper, we provide an O(npolylogn)O(n \mathrm{polylog} n) bound on the expected complexity of the randomly weighted Voronoi diagram of a set of nn 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

    Bregman Voronoi Diagrams: Properties, Algorithms and Applications

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    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

    A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters

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    In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in the plane, the distance between a point tt and a cluster PP is measured as the maximum distance between tt and any point in PP, and the diagram is defined in a nearest-neighbor sense for the input clusters. In this paper we consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for which the combinatorial complexity of the Hausdorff Voronoi diagram is linear in the total number of points, nn, on the convex hulls of all clusters. We present a randomized incremental construction, based on point location, that computes this diagram in expected O(nlog2n)O(n\log^2{n}) time and expected O(n)O(n) space. Our techniques efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The diagram finds direct applications in VLSI computer-aided design.Comment: arXiv admin note: substantial text overlap with arXiv:1306.583

    Fast and Compact Exact Distance Oracle for Planar Graphs

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    For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n5/3)O(n^{5/3})-space distance oracle which answers exact distance queries in O(logn)O(\log n) time for nn-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space O(n2ϵ)O(n^{2 - \epsilon}) for some constant ϵ>0\epsilon > 0) either required query time polynomial in nn or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any Sn3/2S \ge n^{3/2}, we show how to obtain an SS-space distance oracle that answers queries in time O((n5/2/S3/2)logn)O((n^{5/2}/ S^{3/2}) \log n). This is a polynomial improvement over the previous planar distance oracles with o(n1/4)o(n^{1/4}) query time

    Better Tradeoffs for Exact Distance Oracles in Planar Graphs

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    We present an O(n1.5)O(n^{1.5})-space distance oracle for directed planar graphs that answers distance queries in O(logn)O(\log n) time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses O(n5/3)O(n^{5/3})-space and answers queries in O(logn)O(\log n) time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any S[n,n2]S\in [n,n^2], we show an oracle of size SS that answers queries in O~(max{1,n1.5/S})\tilde O(\max\{1,n^{1.5}/S\}) time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of SS and improves by polynomial factors over all the previously known tradeoffs for the range S[n,n5/3]S \in [n,n^{5/3}]
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