Michael Karasick
Lee R. Nackman
Abstract
Many fundamental tests performed by geometric algorithms can be formulated in terms of finding the sign of a determinant. When these tests are implemented using fixed precision arithmetic such as floating point, they can produce incorrect answers; when they are implemented using arbitrary-precision arithmetic, they are expensive to compute. We present adaptive-precision algorithms for finding the signs of determinants of matrices with integer and rational elements. These algorithms were developed and tested by integrating them into the Guibas-Stolfi Delaunay triangulation algorithm. Through a combination of algorithm design and careful engineering of the implementation, the resulting program can triangulate a set of random rational points in the unit circle only four to five times slower than can a floating-point implementation of the algorithm. The algorithms, engineering process, and software tools developed are described.
- 1 AHo, A. V., HOPCRO~, J. E., ANO ULLMAN, J. D. The Design and Analysis o{ Computer Algorithms. Addison-Wesley, Reading, Mass., 1974. Google Scholar
- 2 BENTLEY, J.L. Writing Efficient Programs. Prentice-Hall, Englewood Cliffs, N.J., 1982. Google Scholar
- 3 BERN, M. W., KARLOFF, H. J., RAi;HAVAN, P., ANO SCHEIBER, B. Fast geometric approximation techniques and geometric embedding problems. In Proceedings of the Fifth Annual ACM Symposium on Computational Geometry (June 1989). ACM, New York, 1989, 292-301. Google Scholar
- 4 DOBKIN, D. P., AND SILVER, D. Recipes for geometry and numerical analysis--Part I: An empirical study. In Proceedings of the Fourth Annual ACM Symposium on Computational Geometry (June 1988). ACM, New York, 1988, 93-105. Google Scholar
- 5 EDELSBRUNNER, H., AND M1DCItE, E. P. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. In Proceedings o{ the Fourth Annual ACM Symposium on Computational Geometry (June 1988). ACM, New York, 1988, 118-133. Google Scholar
- 6 FaROOKI, R.T. Numerical stability in geometric algorithms and representations. In Proceedings Mathematics o! Surfaces Ill, D. C. Hanscomb, Ed. Oxford University Press, New York, 1989. Google Scholar
- 7 FORREST, A. R. Computational geometry and software engineering. Towards a geometric computing environment. In Techniques for Computer Graphics, R. A. Earnshaw and D. F. Rogers, Eds. Springer Verlag, New York, 1987, 23-37.Google Scholar
- 8 FORTUNE, S.J. Stable maintenance of point-set triangulations in two dimensions. In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (Oct. 1989). IEEE, New York, 1989.Google Scholar
- 9 GREENE, D. H., AND YAo, F. F, Finite-resolution computational geometry. In Proceedings of the 27th IEEE Symposium on the Foundations of Computer Science (1986). IEEE, New York, 1986, 143-152.Google Scholar
- 10 GUIBA,S, L., SaLESIN, D., aND STOLF!, J. Epsilon geometry: Building robust algorithms from imprecise computations. In Proceedings of the Filth Annual ACM Symposium on Computational Geometry (June 1989). ACM, New York, 1989, 2/)8-217. Google Scholar
- 11 GUIS&S, L. aND STOLFL J. Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams. ACM Trans. Graph. 5, 2 (Apr. 1985), 74-123. Google Scholar
- 12 HOFFMANN, C., HOPCROP'r, J., AND KAR~tSlC{~, M. Towards implementing robust geometric computations. In Proceedings of the Fourth Annual ACM Symposium on Computational Geometry (June 1988). ACM, New York, 1988, 106-117. Google Scholar
- 13 HOFFMANN, C.M. The problem of accuracy and robustness in geometric computation. IEEE Comput. 22 (1989), 31-42. Google Scholar
- 14 KARASlCK, M. On the representation and manipulation of rigid solids, Ph.D. thesis, McGill Univ., Montreal, 1988. Google Scholar
- 15 KNUTH, D.E. Seminurnerical Algorithms. Vol. 2. The Art of Computer Programming. Addison- Wesley, Reading, Mass., 1973. Google Scholar
- 16 LovAsz, L. An Algorithmic Theory o{ Numbers, Graphs and Convexity. Vol. 50. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, 1986.Google Scholar
- 17 MILENKOVIC, V.J. Verifiable implementations of geometric algorithms using finite precision arithmetic. Artif IntelL 37 (Dec. 1988), 377-401. Google Scholar
- 18 MILENKOV1C, V.J. Calculating approximate curve arrangements using rounded arithmetic. In Proceedings o{ the Filth Annual ACM Symposium on Computational Geometry (June 1989). ACM, New York, 1989, 197-207. Google Scholar
- 19 MUDUR, S. P., AND KOPARKAR, P.A. Interval methods for processing geometric objects. IEEE Comput. Graph. Appl. 4, 2 (Feb. 1984), 7-17.Google Scholar
- 20 NERINC, E.D. Linear Algebra and Matrix Theory. John Wiley, New York, 1970.Google Scholar
- 21 OTTMANN, T., THIEMT, G., AND ULLRICH, C. Numerical stability of geometric algorithms. In Proceedings of the Third Annual ACM Symposium on Computational Geometry (June 1987). ACM, New York, 1987, 119-125. Google Scholar
- 22 PREPARATA, F. P., AND VUILLEMIN, J.E. Practical cellular dividers. INRIA Tech. Pep. 807, Rocquencourt, France, March 1988.Google Scholar
- 23 SEGAL, M,, AND SI~QUIN, C. Partitioning polyhedral objects into nonintersecting parts. IEEE Coraput. Graph. Appl. 8, 1 (Jan. 1988), 53-67. Google Scholar
- 24 STROUSTRUP, B. The C*+ Programming Language. Addison-Wesley, Reading, Mass., 1987. Google Scholar
- 25 STROUSTRUP, B. Parameterized types for C*+. In Proceedings of the USENIX C*+ Con{erence (Oct. 1988). USENIX Association.Google Scholar
- 26 SUGIHARA, K. On finite-precision representations of geometric objects. J. Comput. Syst. Sci. 39 (1989), 236-247. Google Scholar
- 27 SUGIHARA, g., AND IRl, M. Construction of the Voronoi diagram for one million generators in single-precision arithmetic. Paper presented at the First Canadian Conference on Computational Geometry. Montreal, Aug. 1989.Google Scholar
- 28 YAP, C. A geometric consistency theorem for a symbolic perturbation scheme. In Proceedings o{ the Fourth Annual ACM Symposium on Computational Geometry (June 1988). ACM, New York, 1988, 134-142. Google Scholar
Index Terms
- Efficient Delaunay triangulation using rational arithmetic
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