Abstract
The combination of divide-and-conquer and random sampling has proven very effective in the design of fast geometric algorithms. A flurry of efficient probabilistic algorithms have been recently discovered, based on this happy marriage. We show that all those algorithms can be derandomized with only polynomial overhead. In the process we establish results of independent interest concerning the covering of hypergraphs and we improve on various probabilistic bounds in geometric complexity. For example, givenn hyperplanes ind-space and any integerr large enough, we show how to compute, in polynomial time, a simplicial packing of sizeO(r d) which coversd-space, each of whose simplices intersectsO(n/r) hyperplanes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P.Aggarwal, and M.Sharir: Red-blue intersection detection algorithms with applications to motion planning and collision detection,Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 70–80.
N. Alon, L. Babai, andA. Itai: A fast and simple randomized parallel algorithm for the maximal independent set problem,J. of Alg.,7 (1986), 567–583.
K. L. Clarkson: A randomized algorithm for closest-point queries,SIAM J. Comput.,17 (1988), 838–847.
K. L. Clarkson: New applications of random sampling in computational geometry,Disc. Comp. Geom.,2 (1987), 195–222.
K. L.Clarkson:Applications of random sampling in computational geometry, II, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 1–11.
K. L.Clarkson, and P. W.Shor:Algorithms for diametral pairs and convex hulls that are optimal, randomized, and incremental, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 12–17.
K. L.Clarkson, R. E.Tarjan, and C. J.Van Wyk:A fast Las Vegas algorithm for triangulating a simple polygon, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 18–22.
H. Edelsbrunner:Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, Germany,1987.
H.Edelsbrunner, L. J.Guibas, J.Hersberger, R.Seidel, M.Sharir, J.Snoeyink, and E.Welzl:Implicitly representing arrangements of lines or segments, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 56–69.
H.Edelsbrunner, L. J.Guibas, and M.Sharir:The complexity of many faces in arrangements of lines and of segments, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 44–55.
H. Edelsbrunner, J. O'Rourke, andR. Seidel: Constructing arrangements of lines and hyperplanes with applications,SIAM J. Comput.,15 (1986), 341–363.
P. Erdős, andJ. Spencer:Probabilistic methods in combinatorics, Academic Press, New York,1974.
D. Haussler, andE. Welzl: Epsilon-nets and simplex range queries,Disc. Comp. Geom.,2 (1987), 127–151.
A. Joffe: On a set of almost deterministick-independent random variables,Ann. of Prob.,2 (1974), 161–162.
L. Lovász: On the ratio of optimal integral and fractional covers,Discrete Math.,13 (1975), 383–390.
L.Lovász:Combinatorial problems and exercises, North-Holland,1979.
J.Pach:Private communication,1988.
J.Pach, and J.Spencer:Explicit codes with low covering radius, IEEE Trans. Information Theory,to appear.
P.Raghavan:Probabilistic construction of deterministic algorithms: approximating packing integer programs, Proc. 27th Annu. IEEE Symp. on Foundat. of Comput. Sci., (1986), 10–18.
J. H.Reif, and S.Sen:Optimal randomized parallel algorithms for computational geometry, Proc. 16th Internat. Conf. Parallel Processing, St. Charles, IL, 1987. Full version, Duke Univ. Tech. Rept., CS-88-01,1988.
J. Spencer: Puncture sets,J. Combinat. Theory A,17 (1974), 329–336.
J.Spencer:Ten lectures on the probabilistic method, CBMS-NSF, SIAM,1987.
R. E.Tarjan, and C. K.Van Wyk: An O(n log logn)-time algorithm for triangulating a simple polygon,SIAM J. Comput. (1988).
N. Sauer: On the density of families of sets,J. Combinat. Theory A,13 (1972), 145–147.
V. N. Vapnik, andA. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl.,16 (1971), 264–280.
E.Welzl:Partition trees for triangle counting and other range searching problems, Proc. 4th Ann. ACM Sympos. Comput. Geom., (1988), 23–33.
Author information
Authors and Affiliations
Additional information
Bernard Chazelle wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8700917. Joel Friedman wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and this Office of Naval Research under Grant N00014-87-K-0467.
A preliminary version of this work has appeared in the proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (1988). 539–549.