A deterministic view of random sampling and its use in geometry

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.

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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.

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Chazelle, B., Friedman, J. A deterministic view of random sampling and its use in geometry. Combinatorica 10, 229–249 (1990). https://doi.org/10.1007/BF02122778

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AMS subject classification (1980)

  • 68 C 25
  • 52 A 22