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Conference on Algorithms and Discrete Applied Mathematics

CALDAM 2017: Algorithms and Discrete Applied Mathematics pp 24–36Cite as

Voronoi Diagram for Convex Polygonal Sites with Convex Polygon-Offset Distance Function

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

Abstract

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearest- and farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-sided convex polygon, and study the nearest- and farthest-site Voronoi diagrams for sets of line segments and convex polygons. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n disjoint line segments is O(nm), which is asymptotically equal to that of the Voronoi diagram of n point sites with respect to the same distance function. In addition, we generalize this result to the Voronoi diagram of disjoint convex polygonal sites. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n convex polygonal sites, each having at most k sides, is \(O(n(m+k))\). Finally, we show that the corresponding farthest-site Voronoi diagram is a tree-like structure with the same combinatorial complexity.

M. De—Supported by DST-INSPIRE Faculty Grant (DST-IFA14-ENG-75).

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Notes

  1. 1.

    A disadvantage of this approach is that relabeling of the input sites will change the diagram. One can adopt the rule of Klein and Wood [12], who break ties by the lexicographic order of the input points, but with such a solution, the Voronoi diagram will not be invariant under rotation of the plane.

References

  1. Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment Voronoi diagrams. Inf. Process. Lett. 100(6), 220–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aurenhammer, F., Klein, R., Lee, D.-T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore (2013)

    Book  MATH  Google Scholar 

  3. Barequet, G., Dickerson, M.T., Goodrich, M.T.: Voronoi diagrams for convex polygon-offset distance functions. Discret. Comput. Geom. 25(2), 271–291 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bohler, C., Cheilaris, P., Klein, R., Liu, C.-H., Papadopoulou, E., Zavershynskyi, M.: On the complexity of higher order abstract Voronoi diagrams. Comput. Geom. Theory Appl. 48(8), 539–551 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.-S.: Farthest-polygon Voronoi diagrams. Comput. Geom. Theory Appl. 44(4), 234–247 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chew, L.P., Drysdale III, R.L.S.: Voronoi diagrams based on convex distance functions. In: O’Rourke, J. (ed.) Proceedings of the First Annual Symposium on Computational Geometry, Baltimore, Maryland, USA, 5–7 June 1985, pp. 235–244. ACM (1985)

    Google Scholar 

  7. Descartes, R.: Principia Philosophiae. Ludovicus Elzevirius, Amsterdam (1644)

    Google Scholar 

  8. Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153–174 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kelley, J.L., Namioka, I.: Linear Topological Spaces. Springer, Heidelberg (1976)

    MATH  Google Scholar 

  10. Kirkpatrick, D.G.: Efficient computation of continuous skeletons. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 29–31 October 1979, pp. 18–27. IEEE Computer Society (1979)

    Google Scholar 

  11. Klein, R.: Concrete and Abstract Voronoi Diagrams. Lecture Notes in Computer Science, vol. 400. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  12. Klein, R., Wood, D.: Voronoi diagrams based on general metrics in the plane. In: Cori, R., Wirsing, M. (eds.) STACS 1988. LNCS, vol. 294, pp. 281–291. Springer, Heidelberg (1988). doi:10.1007/BFb0035852

    Chapter  Google Scholar 

  13. Leven, D., Sharir, M.: Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams. Discret. Comput. Geom. 2, 9–31 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. McAllister, M., Kirkpatrick, D.G., Snoeyink, J.: A compact piecewise-linear Voronoi diagram for convex sites in the plane. Discret. Comput. Geom. 15(1), 73–105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mehlhorn, K., Meiser, S., Rasch, R.: Furthest site abstract Voronoi diagrams. Int. J. Comput. Geom. Appl. 11(6), 583–616 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Papadopoulou, E., Dey, S.K.: On the farthest line-segment Voronoi diagram. Int. J. Comput. Geometry Appl. 23(6), 443–460 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yap, C.-K.: An O\((n \log n)\) algorithm for the Voronoi diagram of a set of simple curve segments. Discret. Comput. Geom. 2, 365–393 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, The Technion—Israel Institute of Technology, Haifa, Israel

    Gill Barequet

  2. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

    Minati De

Authors
  1. Gill Barequet
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  2. Minati De
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Correspondence to Minati De .

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Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

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Barequet, G., De, M. (2017). Voronoi Diagram for Convex Polygonal Sites with Convex Polygon-Offset Distance Function. In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-53007-9_3

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  • Online ISBN: 978-3-319-53007-9

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