Abstract
In a paper that considered arithmetic precision as a limited resource in the design and analysis of algorithms, Liotta, Preparata and Tamassia defined an “implicit Voronoi diagram” supporting logarithmic-time proximity queries using predicates of twice the precision of the input and query coordinates. They reported, however, that computing this diagram uses five times the input precision. We define a reduced-precision Voronoi diagram that similarly supports proximity queries, and describe a randomized incremental construction using only three times the input precision. The expected construction time is O(n (logn + logμ)), where μ is the length of the longest Voronoi edge; we can construct the implicit Voronoi from the reduced-precision Voronoi in linear time.
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.
References
Liotta, G., Preparata, F.P., Tamassia, R.: Robust proximity queries: An illustration of degree-driven algorithm design. SIAM J. Comput. 28(3), 864–889 (1999)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, New York (2008)
Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153–174 (1987)
Guibas, L.J., Knuth, D.E., Sharir, M.: Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7, 381–413 (1992)
Shamos, M.I., Hoey, D.: Closest-point problems. In: SFCS 1975: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, Washington, DC, USA, pp. 151–162. IEEE Computer Society, Los Alamitos (1975)
Karamcheti, V., Li, C., Pechtchanski, I., Yap, C.: A core library for robust numeric and geometric computation. In: SCG 1999: Proceedings of the fifteenth annual symposium on Computational geometry, pp. 351–359 (1999)
Yap, C.K.: Towards exact geometric computation. Comput. Geom. Theory Appl. 7(1-2), 3–23 (1997)
Cgal, Computational Geometry Algorithms Library, http://www.cgal.org
Burnikel, C., Könemann, J., Mehlhorn, K., Näher, S., Schirra, S., Uhrig, C.: Exact geometric computation in LEDA. In: SCG 1995: Proceedings of the eleventh annual symposium on Computational geometry, pp. 418–419 (1995)
Devillers, O., Preparata, F.P.: A probabilistic analysis of the power of arithmetic filters. Discrete and Computational Geometry 20(4), 523–547 (1998)
Devillers, O., Preparata, F.P.: Further results on arithmetic filters for geometric predicates. Comput. Geom. Theory Appl. 13(2), 141–148 (1999)
Fortune, S., Van Wyk, C.J.: Efficient exact arithmetic for computational geometry. In: SCG 1995: Proceedings of the eleventh annual symposium on Computational geometry, pp. 163–172 (1993)
Shewchuk, J.R.: Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates. Discrete & Computational Geometry 18(3), 305–363 (1997)
Sugihara, K., Iri, M.: Construction of the Voronoi diagram for ‘one million’ generators in single-precision arithmetic. Proceedings of the IEEE 80(9), 1471–1484 (1992)
Boissonnat, J.D., Preparata, F.P.: Robust plane sweep for intersecting segments. SIAM J. Comput. 29(5), 1401–1421 (2000)
Boissonnat, J.D., Snoeyink, J.: Efficient algorithms for line and curve segment intersection using restricted predicates. In: SCG 1999: Proceedings of the fifteenth annual symposium on Computational geometry, pp. 370–379 (1999)
Chan, T.M.: Reporting curve segment intersections using restricted predicates. Comput. Geom. Theory Appl. 16(4), 245–256 (2000)
Aurenhammer, F.: Voronoi diagrams a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)
Hoff, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized voronoi diagrams using graphics hardware. In: SIGGRAPH 1999: Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pp. 277–286 (1999)
Breu, H., Gil, J., Kirkpatrick, D., Werman, M.: Linear time Euclidean distance transform algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 17, 529–533 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Millman, D.L., Snoeyink, J. (2009). Computing the Implicit Voronoi Diagram in Triple Precision. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_43
Download citation
DOI: https://doi.org/10.1007/978-3-642-03367-4_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
eBook Packages: Computer ScienceComputer Science (R0)