Abstract
Given a tesselation of the plane, defined by a planar straight-line graph G, we want to find a minimal set S of points in the plane, such that the Voronoi diagram associated with S ‘fits’ G. This is the Generalized Inverse Voronoi Problem (GIVP), defined in [12] and rediscovered recently in [3]. Here we give an algorithm that solves this problem with a number of points that is linear in the size of G, assuming that the smallest angle in G is constant.
Mathematics Subject Classification: 52C45, 65D18, 68U05.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ash, P., Bolker, E.D.: Recognizing Dirichlet Tesselations. Geometriae Dedicata 19, 175–206 (1985)
Aurenhammer, F.: Recognising Polytopical Cell Complexes and Constructing Projection Polyhedra. J. Symbolic Computation 3, 249–255 (1987)
Banerjee, S., Bhattacharya, B.B., Das, S., Karmakar, A., Maheshwari, A., Roy, S.: On the Construction of a Generalized Voronoi Inverse of a Rectangular Tesselation. In: Procs. 9th Int. IEEE Symp. on Voronoi Diagrams in Science and Engineering, pp. 132–137. IEEE, New Brunswick (2012)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry. Algorithms and Applications, 3rd edn. Springer, Berlin (2008)
Hartvigsen, D.: Recognizing Voronoi Diagrams with Linear Programming. ORSA J. Comput. 4, 369–374 (1992)
Martínez, A., Martínez, J., Pérez-Rosés, H., Quirós, R.: Image Processing using Voronoi diagrams. In: Procs. 2007 Int. Conf. on Image Proc., Comp. Vision, and Pat. Rec., pp. 485–491. CSREA Press (2007)
Ratti, B., Sommer, C.: Approximating Shortest Paths in Spatial Social Networks. In: Procs. 2012 ASE/IEEE Int. Conf. on Social Computing and 2012 ASE/IEEE Int. Conf. on Privacy, Security, Risk and Trust, pp. 585–586. IEEE Comp. Soc. (2012)
Schoenberg, F.P., Ferguson, T., Li, C.: Inverting Dirichlet Tesselations. The Computer J. 46, 76–83 (2003)
Sommer, C.: Approximate Shortest Path and Distance Queries in Networks. PhD Thesis, Department of Computer Science, The University of Tokyo, Japan (2010)
Sugihara, K., Iri, M.: Construction of the Voronoi Diagram for ‘One Million’ Generators in Single-Precision Arithmetic. Procs. IEEE 80, 1471–1484 (1992)
Trinchet-Almaguer, D.: Algorithm for Solving the Generalized Inverse Voronoi Problem. Honour’s Thesis, Department of Computer Science, University of Oriente, Cuba (2005) (in Spanish)
Trinchet-Almaguer, D., Pérez-Rosés, H.: Algorithm for Solving the Generalized Inverse Voronoi Problem (in Spanish). Revista Cubana de Ciencias Informaticas 1(4), 58–71 (2007)
Yeganova, L., Falk, J.E., Dandurova, Y.V.: Robust Separation of Multiple Sets. Nonlinear Analysis 47, 1845–1856 (2001)
Yeganova, L.E.: Robust linear separation of multiple finite sets. Ph.D. Thesis, George Washington University (2001)
Zhou, B., Pei, J., Luk, W.-S.: A brief survey on anonymization techniques for privacy preserving publishing of social network data. ACM SIGKDD Explorations Newsletter 10, 12–22 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aloupis, G., Pérez-Rosés, H., Pineda-Villavicencio, G., Taslakian, P., Trinchet-Almaguer, D. (2013). Fitting Voronoi Diagrams to Planar Tesselations. In: Lecroq, T., Mouchard, L. (eds) Combinatorial Algorithms. IWOCA 2013. Lecture Notes in Computer Science, vol 8288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45278-9_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-45278-9_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45277-2
Online ISBN: 978-3-642-45278-9
eBook Packages: Computer ScienceComputer Science (R0)