Summary
In this chapter, we address the problem of robust point-location in a generalized d-dimensional Voronoi diagram. The exact point location requires the solution for expressions of degree four. A natural question is what can be done using expression of smaller degree. We apply polyhedral metrics for this task. In general dimensions two Minkowski metrics can be used: L 1 (Manhattan metric) and L ∞ (supremum metric). The approximation factor is \(\sqrt d\) and the computation uses expressions of degree one.
We also show that a polygonal metric can be applied in two dimensions. The computation involves only \(O(\lg k)\) calls of the algorithm ESSA for detecting the sign of a sum using floating-point arithmetic.
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References
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer, Heidelberg (1997)
Burnikel, C.: Exact Computation of Voronoi Diagrams and Line Segment Intersections. Ph.D thesis, Universität des Saarlandes (March 1996)
Chen, Z., Papadopoulou, E., Xu, J.: Robustness of algorithm for k-gon Voronoi diagram construction. Information Processing Letters 97(4), 138–145 (2006)
Chew, L.P., Drysdale III, R.L.: Voronoi diagrams based on convex distance functions. In: Proc. 1st Annu. ACM Sympos. Comput. Geom., pp. 235–244 (1985)
Dey, T.K., Sugihara, K., Bajaj, C.L.: Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design 9, 457–470 (1992)
Edelsbrunner, H., Mücke, E.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. In: Proc. 4th Annu. ACM Sympos. Comput. Geom., pp. 118–133 (1988)
Fortune, S., Wyk, C.: Efficient exact arithmetic for computational geometry. In: Proc. 9th Annu. ACM Sympos. Comput. Geom., pp. 163–172 (1993)
Gavrilova, M.: A reliable algorithm for computing the generalized Voronoi diagram for a set of spheres in the Euclidean d-dimensional space. In: Canadian Conference on Computational Geometry, pp. 82–87 (2002)
Gavrilova, M., Bespamyatnikh, S.: On exact solution of a point-location problem in a system of d-dimensional hyperbolic surfaces. In: Canadian Conference on Computational Geometry, pp. 136–139 (2003)
Gavrilova, M., Ratschek, H., Rokne, J.: Exact computation of Voronoi diagram and Delaunay triangulation. Reliable Computing 6(1), 39–60 (2000)
Gavrilova, M.: Algorithm library development for complex biological and mechanical systems. In: DIMACS Workshop on Implementation of Geometric Algorithms (2002)
Hoffmann, C.M., Hopcroft, J.E., Karasik, M.S.: Towards implementing robust geometric computations. In: Proc. 4th Annu. ACM Sympos. Comput. Geom., pp. 106–117 (1988)
Krishnan, S., Foskey, M., Culver, T., Keyser, J., Manocha, D.: PRECISE: Efficient multiprecision evaluation of algebraic roots and predicates for reliable geometric computations. In: Proc. 17th Annu. ACM Sympos. Comput. Geom., pp. 274–283 (2001)
Liotta, G., Preparata, F.P., Tamassia, R.: Robust proximity queries: an illustration of degree-driven algorithm design. SIAM J. Comput. 28(3), 864–889 (1998)
McAllister, M., Kirkpatrick, D., Snoeyink, J.: A compact piecewise-linear Voronoi diagram for convex sites in the plane. Discrete Comput. Geom. 15, 73–105 (1996)
S. Naher, The LEDA user manual, Version 3.1 (January 16, 1995), ftp.mpi-sb.mpg.de
Stewart, A.J.: Robust point location in approximate polygons.In: Canadian Conference on Computational Geometry, pp. 179–182 (1991)
Sugihara, K., Iri, M.: A robust topology-oriented incremental algorithm for Voronoi diagrams. International Journal of Computational Geometry and Applications 4(2), 179–228 (1994)
Widmayer, P., Wu, Y.F., Wong, C.K.: On some distance problems in fixed orientations. SIAM J. Comput. 16, 728–746 (1987)
Yap, C., Dube, T.: The exact computation paradigm. In: Du, D.-Z., Hwang, F.K. (eds.) Computing in Euclidean Geometry, 2nd edn., World Scientific Press, Singapore (1995)
Yap, C.K.: Symbolic treatment of geometric degeneracies. Journal of Symbolic Computation 10, 349–370 (1990)
Yap, C.K.: Toward exact geometric computation. Computational Geometry: Theory and Applications 7, 3–23 (1997)
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Bereg, S., Gavrilova, M.L., Zhang, Y. (2009). Robust Point-Location in Generalized Voronoi Diagrams. In: Gavrilova, M.L. (eds) Generalized Voronoi Diagram: A Geometry-Based Approach to Computational Intelligence. Studies in Computational Intelligence, vol 158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85126-4_13
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DOI: https://doi.org/10.1007/978-3-540-85126-4_13
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