Abstract Voronoi diagrams are a unifying framework that covers many types of concrete Voronoi diagrams. This talk reports on the state of the art, including recent progress.European Science Foundatio

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Klein, Rolf

Márquez Pérez, Alberto (Coordinador)

Urrutia Galicia, Jorge (Coordinador)

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Abstract Voronoi diagramsRolf Klein∗Universität Bonn, Institut für Informatik IAbstractAbstract Voronoi diagrams are a unifying frameworkthat covers many types of concrete Voronoi diagrams.This talk reports on the state of the art, includingrecent progress.IntroductionConcrete Voronoi diagrams [1] are mostly dened interms of sites and distance, and both concepts canvary greatly. Abstract Voronoi diagrams [6] are builton what most concrete diagrams have in common: asystem of simple bisecting curves J(p, q) = J(q, p),where p, q are just indices from a set S of n elements.Each curve J(p, q) divides the plane into two domains,D(p, q) and D(q, p). The abstract Voronoi region of pwith respect to S is dened byVR(p, S) :=⋂q∈S\{p}D(p, q)and the abstract Voronoi diagram of S is just the planeminus all Voronoi regions.An interesting question is what properties to re-quire of the curves J(p, q). They should be as weakas possible for generality, but strong enough to en-sure that useful Voronoi structures result from theabove denitions. It turns out [7] that the followingare sucient.(A1) Each curve J(p, q), where p 6= q, is unbounded.After stereographic projection to the sphere,it can be completed to a closed Jordan curvethrough the north pole.For any three indices p, q, r in S, and S′ :={p, q, r},(A2) each Voronoi region VR(p, S′) is path-wise con-nected,(A3) each point of the plane belongs to the closure ofa Voronoi region VR(p, S′).∗Email: rolf.klein@uni-bonn.de. This work was supportedby the European Science Foundation (ESF) in the EURO-CORES collaborative research project EuroGIGA/VORONOI.p rpqqrp rpq q rpqprrqprp qq rFigure 1: Admissible curve systemsInformally, if the bisecting curves are un-bounded and behave decently, and if any tripletJ(p, q), J(p, r), J(q, r) is situated as shown in Fig-ure 1, the AVD theory applies.1 ResultsThis means that structural results and ecient algo-rithms become available without further eort [7].Theorem 1 V (S) is a planar graph of complexityO(n). It can be constructed in an expected numberof O(n log n) many steps.If we replace Axiom A2 by the more general re-quirement(A2') Each Voronoi region VR(p, S′) has at most s con-nected components(and assume that any two curves intersect onlynitely often), the above result can be generalized asfollows [3].Theorem 2 Abstract Voronoi diagrams with discon-nected regions can be computed in an expected numberofOs2n n∑j=3mjjsteps, where mj denotes the average number of facesper region in all AVDs of j sites from S.55Abstract Voronoi diagramsOne can extend the denition of abstract Voronoidiagrams to orders k > 1 by deningVRk(P, S) :=⋂p∈P, q∈S\PD(p, q).For order k = n−1, the resulting AVDs are trees [9] oflinear size. In the general case the following complex-ity result holds. Here we assume that all curves arein general position, and that the standard Voronoi-regions are non-empty.Theorem 3 The abstract order-k Voronoi diagramV k(S) has at most 2k(n− k) many faces. This boundcan be achieved.2 ConclusionOpen is the case of closed bisecting curves.References[1] F. Aurenhammer, R. Klein, and D.-T. Lee, VoronoiDiagrams and Delaunay Triangulations, World Sci-entic Publishing Company, to appear August 2013.[2] C. Bohler, P. Cheilaris, R. Klein, C.H. Liu, E. Pa-padopoulou, and M. Zavershynskyi, On the complex-ity of higher order abstract Voronoi diagrams, to bepresented at ICALP' 13.[3] C. Bohler and R. Klein, Abstract Voronoi diagramswith disconnected regions, Bonn 2013, manuscript.[4] C. Bohler and R. Klein, Point sites with individualdistance functions, Bonn 2012, manuscript.[5] A. G. Corbalan, M. Mazon, and T. Recio, Geometryof bisectors for strictly convex distance functions, In-ternational Journal of Computational Geometry andApplications 6(1) (1996), 4558.[6] R. Klein, Concrete and abstract Voronoi diagrams,Lecture Notes in Computer Science, 400, Springer-Verlag, 1987.[7] R. Klein, E. Langetepe, and Z. Nilforoushan, Ab-stract Voronoi diagrams revisited, ComputationalGeometry: Theory and Applications 42(9) (2009),885902.[8] R. Klein, K. Mehlhorn, and S. Meiser, Random-ized incremental construction of abstract Voronoi di-agrams, Computational Geometry: Theory and Ap-plications 3 (1993), 157184.[9] K. Mehlhorn, S. Meiser, and R. Rasch, Furthestsite abstract Voronoi diagrams, International Journalof Computational Geometry and Applications 11(6)(2001), 583616.56

Abstract Voronoi diagrams

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Abstract Voronoi diagrams

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