New Applications of random Sampling in Computational Geometry

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requires O(s) expected preprocessing time to build a search structure for an arrangement of s hyperplanes in d dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query point p, the cell of the arrangement containing p can be found in O(log s) worst-case time. (The bound holds for any fixed ǫ > 0, with the constant factors dependent on d and ǫ.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expected O(n) time, where n is the total number of vertices of the two polytopes. This matches previous results [11] for the case d = 3 and extends them. Another algorithm samples points in the plane to determine their order k Voronoi diagram, and requires expected O(sk) time for s points. (It is assumed that no four of the points are cocircular.) This sharpens the bound O(sk log s) for Lee’s algorithm [21], and O(s log s+ k(s− k) log s) for Chazelle and Edelsbrunner’s algorithm [4]. Finally, random sampling is used to show that any set of s points in E has O(sk log s/(log log s)) distinct j-sets with j ≤ k. (For S ⊂ E, a set S ⊂ S with |S| = j is a j-set of S if there is a halfspace h with S = S∩h.) This sharpens with respect to k the previous bound O(sk) [5]. The proof of the bound given here is an instance of a “probabilistic method” [16]. A preliminary version of this paper appeared in the Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 1986.