Abstract
In Geographic Information Systems the reversibility of map update operations has not been explored yet. In this paper we are using the Voronoi based Quad-edge data structure to define reversible map update operations. The reversibility of the map operations has been formalised at the lowest level, as the basic algorithms for addition, deletion and moving of spatial objects. Having developed reversible map operations on the lowest level, we were able to maintain reversibility of the map updates at higher levels as well. The reversibility in GIS can be used for efficient implementation of rollback mechanisms and dynamic map visualisations. In order to use the reversibility within the kinetic Voronoi diagram of points and open oriented line segments, we need to assure that reversing the map commands will produce exactly the changes in the map equivalent to the previous map states. To prove that reversing the map update operations produces the exact reverse changes, we show an isomorphism between the set of complex operations on the kinetic Voronoi diagram of points and open oriented line segments and the sets of numbers of new / deleted Voronoi regions induced by these operations, and its explanation using the finite field of residual classes of integers modulo 5: F 5 = ℤ/5ℤ. We show also an isomorphism between the set of complex operations on the kinetic Voronoi diagram of points and open oriented line segments and the set of differences of new and deleted Quad-Edge edges induced by these operations, and its explanation using the commutative ring ℤ15 = ℤ/15ℤ. We show finally the application of these theoretical results to the logging of a kinetic line Voronoi data structure.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anton, F., Gold, C.M.: An iterative algorithm for the determination of Voronoi vertices in polygonal and non-polygonal domains. In: Proceedings of the 9th Canadian Conference on Computational Geometry (CCCG 1997), Kingston, Canada, pp. 257–262 (1997)
Bellman, R.: On a class of functional equations of modular type. Proc. Nat. Acad. Sci. U.S.A. 42, 626–629 (1956)
Cui, J.: A decoding algorithm for general \(\Bbb Z\sb 4\)-linear codes. J. Syst. Sci. Complex. 17(1), 16–22 (2004)
Frank, M., Knight, T., Margolus, N.: Reversibility in optimally scalable computer architectures. In: The First International Conference on Unconventional Models of Computation, January 1998, pp. 165–182 (1998)
Gold, C.M.: Space revisited - back to the basics. In: Proceedings of the Fourth International Symposium on Spatial Data Handling, Zurich, Switzerland, pp. 175–189 (1990)
Gold, C.M.: An object-based dynamic spatial data model, and its applications in the development of a user-friendly digitizing system. In: Proceedings of the Fifth International Symposium on Spatial Data Handling, Charleston, pp. 495–504 (1992)
Gold, C.M.: Three approaches to automated topology, and how computational geometry helps. In: Proceedings of the Sixth International Seminar on Spatial Data Handling, Edinburgh, Scotland, pp. 145–158 (1994)
Gold, C.M., Dakowicz, M.: Kinetic Voronoi/Delaunay Drawing Tools. In: ISVD 2006, pp. 76–84 (2006)
Gold, C.M., Remmele, P.R., Roos, T.: Voronoi Diagrams of Line Segments Made Easy. In: Proceedings of the Seventh Canadian Conference in Computational Geometry (CCCG 1995), Québec, Canada, pp. 223–228 (1995)
Guibas, L., Stolfi, J.: Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams. ACM Transactions on Graphics 4(2), 74–123 (1985)
Ireland, F.K., Rosen, I.M.: A classical introduction to modern number theory. Graduate Texts in Mathematics, vol. 84. Springer, New York (1982); revised edition of Elements of number theory
Lee, Y., Scheidler, R., Yarrish, C.: Computation of the fundamental units and the regulator of a cyclic cubic function field. Experiment. Math. 12(2), 211–225 (2003)
Mioc, D., Anton, F., Gold, C.M., Moulin, B.: Spatio-temporal change representation and map updates in a dynamic Voronoi data structure. In: Proceedings of the Eight International Symposium on Spatial Data Handling, Vancouver, Canada, pp. 441–452 (1998)
Mioc, D., Anton, F., Gold, C.M., Moulin, B.: ”Time travel” Visualization in a Dynamic Voronoi Data Structure. Cartography and GIS 26(2), 99–108 (1999)
Mioc, D., Anton, F., Gold, C.M., Moulin, B.: Map updates in a dynamic Voronoi data structure. In: ISVD 2006, pp. 264–269 (2006)
Okabe, A., Boots, B., Sugihara, K., Nok Chiu, S.: Spatial tessellations: concepts and applications of Voronoi diagrams, 2nd edn. John Wiley & Sons Ltd., Chichester (2000); With a foreword by D. G. Kendall
Prusinkiewicz, P., Lindenmayer, A.: The Algorithmic Beauty of Plants. Springer, New York (1990)
Reversible Computing FAQ, http://www.cise.ufl.edu/research/revcomp/faq.html
Roos, T.: Dynamic Voronoi diagrams. Ph.D. Thesis, University of Würzburg, Germany (1991)
Scheidler, R., Stein, A.: Voronoi’s algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp. 69(231), 1245–1266 (2000)
Vaario, J.: An Emergent Modeling Method for Artificial Neural Networks. Doctoral dissertation, University of Tokyo, Japan (1993)
Vleugels, J., Kok, J.N., Overmars, M.: Motion Planning with Complete Knowledge using a Colored SOM. International Journal of Neural Systems 8(5-6), 613–628 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mioc, D., Anton, F., Gold, C., Moulin, B. (2010). Kinetic Line Voronoi Operations and Their Reversibility. In: Gavrilova, M.L., Tan, C.J.K., Anton, F. (eds) Transactions on Computational Science IX. Lecture Notes in Computer Science, vol 6290. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16007-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-16007-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16006-6
Online ISBN: 978-3-642-16007-3
eBook Packages: Computer ScienceComputer Science (R0)