Advertisement

On the Construction of Generalized Voronoi Inverse of a Rectangular Tessellation

  • Sandip Banerjee
  • Bhargab B. Bhattacharya
  • Sandip Das
  • Arindam Karmakar
  • Anil Maheshwari
  • Sasanka Roy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8110)

Abstract

We introduce a new concept of constructing a generalized Voronoi inverse (GVI) of a given tessellation \({\mathcal T}\) of the plane. Our objective is to place a set S i of one or more sites in each convex region (cell) \(t_i \in{\mathcal T}\), such that all edges of \({\mathcal T}\) coincide with edges of Voronoi diagram V(S), where S = ∪  i S i , and ∀ i,j, i ≠ j, S i  ∩ S j  = ∅. Computation of GVI in general, is a difficult problem. In this paper, we study properties of GVI for the case when \(\mathcal T\) is a rectangular tessellation and propose an algorithm that finds a minimal set of sites S. We also show that for a general tessellation, a solution of GVI always exists.

Keywords

Heat Sink Voronoi Diagram Junction Point Voronoi Region Perpendicular Bisector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, V.J., Urrutia, G. (eds.) Handbook of Computational Geometry, pp. 201–290. Elsevier Science Publishing (2000)Google Scholar
  2. 2.
    Ash, P., Bolker, E., Crapo, H., Whiteley, W.: Convex polyhedra, Dirichlet tessellations, and spider webs. In: Senechal, M., Fleck, G. (eds.) Shaping Space: A Polyhedral Approach, ch. 17, pp. 231–250. Birkhauser, Basel (1988)Google Scholar
  3. 3.
    Balzer, M., Heck, D.: Capacity-constrained Voronoi diagrams in finite spaces. In: Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering, pp. 44–56 (2008)Google Scholar
  4. 4.
    Balzer, M.: Capacity-constrained Voronoi diagrams in continuous spaces. In: Proceedings of the 5th International Symposium on Voronoi Diagrams in Science and Engineering, pp. 79–88 (2009)Google Scholar
  5. 5.
    Gavrilova, M.L.: Generalized Voronoi Diagram: A Geometry-Based approach to computational intelligence. SCI, vol. 15 (2008)Google Scholar
  6. 6.
    Hartvigsen, D.: Recognizing Voronoi diagrams with linear programming. ORSA J. Comput. 4(4), 369–374 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Suzuki, A., Iri, M.: Approximation of a tessellation of the plane by a Voronoi diagram. J. Oper. Res. Soc. Japan 29, 69–96 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Yuksek, K., Cezayirli, A.: Linking image zones to database by using inverse Voronoi diagrams: A Novel Liz-Ivd Method. In: IEEE International Symposium on Intelligent Control, Saint Petersburg, Russia, July 8-10, pp. 423–427 (2009)Google Scholar
  9. 9.
    Drezner, Z., Hamacher, H.W. (eds.): Facility location: applications and theory. Springer (2002)Google Scholar
  10. 10.
    Hanan, M.: On Steiners problem with rectilinear distance. SIAM Journal Appl. Math 14, 255–265 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goplen, B.: Advanced placement techniques for future VLSI circuits: A short term longitudinal study, University of Minnesota (2006)Google Scholar
  12. 12.
    Tsai, C.H., Kang, S.M.: Cell-Level placement for improving substrate thermal distribution. IEEE Trans. CAD 19(2), 253–266 (2000)CrossRefGoogle Scholar
  13. 13.
    Chen, G., Sapatnekar, S.S.: Partition-driven standard cell thermal placement. In: Proceedings of the International Symposium on Physical Design, pp. 75–80 (2003)Google Scholar
  14. 14.
    Chakrabarty, K., Xu, T.: Digital Microfluidic Biochips: Design and Optimization. CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  15. 15.
    Bishop, C.J.: Non obtuse triangulations of PSLGS (2010) (manuscript )Google Scholar
  16. 16.
    Hangan, T., Itoh, J., Zamfirescu, T.: Acute triangulations. Bull. Math. Soc. Sci. Math. Roumanie 43, 279–286 (2000)MathSciNetGoogle Scholar
  17. 17.
    Yuan, L.: Acute triangulations of polygons. Discrete and Computational Geometry 34(4), 697–706 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Edelsbrunner, H.: Triangulations and meshes in computational geometry. Acta Numerica 9, 133–213 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zamfirescu, C.T.: Survey of two-dimensional acute triangulations. Discrete Mathematics 313(1), 35–49 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Earten, H., Ungor, A.: Computing acute and non obtuse triangulations. In: Canadian Conference on Computational Geometry, Ottawa, Canada (2007)Google Scholar
  21. 21.
    Du, D.Z., Hwang, F.: Mesh generation and optimal triangulation. In: Bern, M., Eppstein, D. (eds.) Computing in Euclidean Geometry, pp. 23–80. World Scientific (1995)Google Scholar
  22. 22.
    Wimer, S., Koren, I., Cederbaum, I.: Optimal aspect ratios of building blocks in VLSI. IEEE Trans. CAD 8(2), 139–145 (1989)CrossRefGoogle Scholar
  23. 23.
    Wang, T.C., Wong, D.F.: Optimal floorplan area optimization. IEEE Trans. CAD 11(8), 992–1002 (1992)CrossRefGoogle Scholar
  24. 24.
    Majumder, S., Sur-Kolay, S., Nandy, S.C., Bhattacharya, B.B., Chakraborty, B.: Hot spots and zones in a chip: A geometrician’s view. In: Poc. Int. Conf. VLSI Design, pp. 691–696 (2005)Google Scholar
  25. 25.
    Majumder, S., Bhattacharya, B.B.: Solving thermal problems of hot chips using Voronoi diagrams. In: Poc. Int. Conf. VLSI Design, pp. 545–548 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sandip Banerjee
    • 1
  • Bhargab B. Bhattacharya
    • 1
  • Sandip Das
    • 1
  • Arindam Karmakar
    • 2
  • Anil Maheshwari
    • 3
  • Sasanka Roy
    • 4
  1. 1.ACM UnitIndian Statistical InstituteKolkataIndia
  2. 2.Tezpur UniversityTezpurIndia
  3. 3.Carleton UniversityOttawaCanada
  4. 4.Chennai Mathematical InstituteChennaiIndia

Personalised recommendations