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Revisiting Hyperbolic Voronoi Diagrams in Two and Higher Dimensions from Theoretical, Applied and Generalized Viewpoints

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Book cover Transactions on Computational Science XIV

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 6970))

Abstract

This paper revisits hyperbolic Voronoi diagrams, which have been investigated since mid 1990’s by Onishi et al., from three standpoints, background theory, new applications, and geometric extensions.

First, we review two ideas to compute hyperbolic Voronoi diagrams of points. One of them is Onishi’s method to compute a hyperbolic Voronoi diagram from a Euclidean Voronoi diagram. The other one is a linearization of hyperbolic Voronoi diagrams. We show that a hyperbolic Voronoi diagram of points in the upper half-space model becomes an affine diagram, which is part of a power diagram in the Euclidean space. This gives another proof of a result obtained by Nielsen and Nock on the hyperbolic Klein model. Furthermore, we consider this linearization from the view point of information geometry. In the parametric space of normal distributions, the hyperbolic Voronoi diagram is induced by the Fisher metric while the divergence diagram is given by the Kullback-Leibler divergence on a dually flat structure. We show that the linearization of hyperbolic Voronoi diagrams is similar to one of two flat coordinates in the dually flat space, and our result is interesting in view of the linearization having information-geometric interpretations.

Secondly, from the viewpoint of new applications, we discuss the relation between the hyperbolic Voronoi diagram and the greedy embedding in the hyperbolic plane. Kleinberg proved that in the hyperbolic plane the greedy routing is always possible. We point out that results of previous studies about the greedy embedding use a property that any tree is realized as a hyperbolic Delaunay graph easily.

Finally, we generalize hyperbolic Voronoi diagrams. We consider hyperbolic Voronoi diagrams of spheres by two measures and hyperbolic Voronoi diagrams of geodesic segments, and propose algorithms for them, whose ideas are similar to those of computing hyperbolic Voronoi diagrams of points.

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References

  1. Alt, H., Cheong, O., Vigneron, A.: The Voronoi diagram of curved objects. Discrete & Computational Geometry 34, 439–453 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amari, S.: α-divergence is unique, belonging to both f-divergence and Bregman divergence classes. IEEE Transactions on Information Theory 55(11), 4925–4931 (2009)

    Article  MathSciNet  Google Scholar 

  3. Amari, S., Nagaoka, H.: Methods of Information Geometry. Oxford University Press (2000)

    Google Scholar 

  4. Aurenhammer, F.: Power diagrams: Properties, algorithms and applications. SIAM Journal of Computing 16(1), 78–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aurenhammer, F.: Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Computing Surveys 23, 345–405 (1991)

    Article  Google Scholar 

  6. Ben-Chen, M., Gotsman, C., Gortler, S.J.: Routing with guaranteed delivery on virtual coordinates. In: Proceedings of the 18th Canadian Conference on Computational Geometry, pp. 117–120 (2006)

    Google Scholar 

  7. Ben-Chen, M., Gotsman, C., Wormser, C.: Distributed computation of virtual coordinates. In: Proceedings of the 23rd Annual Symposium on Computational Geometry (SCG 2007), pp. 210–219 (2007)

    Google Scholar 

  8. Boissonnat, J.-D., Karavelas, M.: On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In: 14th ACM-SIAM Symposium on Discrete Algorithms, pp. 305–312 (2003)

    Google Scholar 

  9. Boissonnat, J.-D., Delage, C.: Convex Hull and Voronoi Diagram of Additively Weighted Points. In: Proceedings of the 13th Annual European Symposium on Algorithms, pp. 367–378 (2005)

    Google Scholar 

  10. Bose, P., Morin, P.: Online Routing in Triangulations. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 113–122. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  11. Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7(3), 200–217 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cannon, J.W., Floyd, W.J., Kenyon, R., Parry, W.R.: Hyperbolic Geometry. Flavors of Geometry MSRI Publications 31, 59–115 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Cvetkovski, A., Crovella, M.: Hyperbolic embedding and routing for dynamic graphs. In: Proceedings of the 28th IEEE Conference on Computer Communications (INFOCOM 2009), pp. 1647–1655 (2009)

    Google Scholar 

  14. Drysdale, R.L., Lee, D.T.: Generalized Voronoi diagrams in the plane. In: Proc. 16th Allerton Conference on Communication, Control, and Computing, pp. 833–842 (1978)

    Google Scholar 

  15. Eppstein, D., Goodrich, M.T.: Succinct Greedy Graph Drawing in the Hyperbolic Plane. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 14–25. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  16. Hayashi, M.: Quantum Information — An Introduction. Springer, Berlin (2006)

    MATH  Google Scholar 

  17. Hayashi, M., Imai, H., Matsumoto, K., Ruskai, M.B., Shimono, T.: Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures. Quantum Information and Computation 5, 13–31 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Hiroshima, T., Miyamoto, Y., Sugihara, K.: Another proof of polynomial-time recognizability of Delaunay graphs. IEICE Trans. Fundamentals E83-A(4), 627–638 (2000)

    Google Scholar 

  19. Hodgson, C.D., Rivin, I., Smith, W.: A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bulletin of the American Mathematical Society 27(2), 246–251 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Imai, H., Iri, M., Murota, K.: Voronoi diagram in the Laguerre geometry and its applications. SIAM Journal on Computing 14, 93–105 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato, K., Oto, M., Imai, H., Imai, K.: Computational geometry analysis of quantum state space and its applications. In: Gavrilova, M.L. (ed.) Generalized Voronoi Diagrams, pp. 67–108. Springer, Berlin (2009)

    Google Scholar 

  22. Kleinberg, R.: Geographic routing using hyperbolic space. In: Proceedings of the 26th Annual IEEE Conference of the IEEE Computer Communications (INFOCOM 2007), pp. 1902–1909 (2007)

    Google Scholar 

  23. Leong, B., Liskov, B., Morris, R.: Greedy virtual coordinates for geographic routing. In: Proceedings of the IEEE International Conference on Network Protocols (ICNP 2007), pp. 71–80 (2007)

    Google Scholar 

  24. Maymounkov, P.: Greedy embeddings, trees, and Euclidean vs. Lobachevsky geometry, Online manuscript, M.I.T. (2006), http://pdos.csail.mit.edu/petar/papers/maymounkov-greedy-prelim.pdf

  25. Nielsen, F., Boissonnat, J.-D., Nock, R.: On Bregman Voronoi diagrams. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete algorithms (SODA 2007), pp. 746–755. SIAM (2007)

    Google Scholar 

  26. Nielsen, F., Nock, R.: Hyperbolic Voronoi diagrams made easy. In: Proceedings of the 2010 International Conference on Computational Science and Its Applications (ICCSA 2010), Fukuoka (2010)

    Google Scholar 

  27. Nilforoushan, Z., Mohades, A.: Hyperbolic Voronoi Diagram. In: Gavrilova, M.L., Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Laganá, A., Mun, Y., Choo, H. (eds.) ICCSA 2006. LNCS, vol. 3984, pp. 735–742. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  28. Nilforoushan, Z., Mohades, A., Rezaii, M.M., Laleh, A.: 3D hyperbolic Voronoi diagrams. Computer-Aided Design 42(9), 759–767 (2010)

    Article  Google Scholar 

  29. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi diagrams, 2nd edn. Wiley Series in Probability and Statistics. Wiley (2000)

    Google Scholar 

  30. Onishi, K.: Riemannian Computational Geometry - Convex Hull, Voronoi Diagram and Delaunay-type Triangulation, Doctoral Thesis, Department of Computer Science, University of Tokyo (1998)

    Google Scholar 

  31. Onishi, K., Takayama, N.: Construction of Voronoi diagram on the upper half-plane. IEICE Trans. Fundamentals E79-A(4), 533–539 (1996)

    Google Scholar 

  32. Onishi, K., Imai, H.: Voronoi diagram in statistical parametric space by Kullback-Leibler divergence. In: Proceedings of the 13th ACM Annual Symposium on Computational Geometry (SCG 1997), pp. 463–465 (1997)

    Google Scholar 

  33. Onishi, K., Imai, H.: Voronoi diagrams for an exponential family of probability distributions in information geometry. In: Proceedings of the Japan-Korea Joint Workshop on Algorithms and Computation, Fukuoka, pp. 1–8 (1997)

    Google Scholar 

  34. Oto, M., Imai, H., Imai, K.: Computational geometry on 1-qubit quantum states. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering, Tokyo, pp. 145–151 (2004)

    Google Scholar 

  35. Papadimitriou, C., Ratajczak, D.: On a conjecture related to geometric routing. Theoretical Computer Science 344(1), 3–14 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rong, G., Jin, M., Guo, X.: Hyperbolic centroidal Voronoi tessellation. In: Proceedings of the 14th ACM Symposium on Solid and Physical Modeling, pp. 117–126 (2010)

    Google Scholar 

  37. Sugihara, K.: Laguerre Voronoi diagram on the sphere. Journal for Geometry and Graphics 6(1), 69–81 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Tanuma, T., Imai, H., Moriyama, S.: Revisiting Hyperbolic Voronoi Diagrams from Theoretical, Applied and Generalized Viewpoints. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering 2010, pp. 23–32 (2010)

    Google Scholar 

  39. Yap, C.K.: An O(n logn) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete & Computational Geometry 2, 365–393 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Information Science and Technology, University of Tokyo, Japan

    Toshihiro Tanuma

  2. CS Dept./CINQIE, University of Tokyo, Japan

    Hiroshi Imai

  3. Graduate School of Information Sciences, Tohoku University, Japan

    Sonoko Moriyama

Authors
  1. Toshihiro Tanuma
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  2. Hiroshi Imai
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  3. Sonoko Moriyama
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. Exascala Ltd., Unit 9, 97 Rickman Drive, B15 2AL, Birmingham, UK

    C. J. Kenneth Tan

  3. Département des sciences géomatiques, Université Laval Centre de Recherche en Géomatique, G1V 0A6, Québec, QC, Canada

    Mir Abolfazl Mostafavi

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Tanuma, T., Imai, H., Moriyama, S. (2011). Revisiting Hyperbolic Voronoi Diagrams in Two and Higher Dimensions from Theoretical, Applied and Generalized Viewpoints. In: Gavrilova, M.L., Tan, C.J.K., Mostafavi, M.A. (eds) Transactions on Computational Science XIV. Lecture Notes in Computer Science, vol 6970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25249-5_1

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  • DOI: https://doi.org/10.1007/978-3-642-25249-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25248-8

  • Online ISBN: 978-3-642-25249-5

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