A Deterministic View of Random Sampling and its Use in Geometry
The combination of divide-and conquer and random sampling has proven very effective in the design of fast geometric algorithms. A flurry of effcient 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, given n hyperplances in d-space and any integer r large enough, we show how to compute, in polynomial
time a simplicial packing of size O(rd) which covers d-space, each of whose simplices intersects O(n/r) hyperplanes. Also, we show how to locate a point among n hyperplanes in d-space in O(log n) query time, using O(nd) storage and polynomial preprocessing.