Separating collections of points in Euclidean spaces

Abstract

Abstract Given two disjoint convex sets A and B in Ed, a hyperplane h in Ed separates them if A lies on one of the half spaces defined by h while B lies on the complementary half space. Given a collection F of convex sets in Ed we say F is separated by a set of hyperplanes H if every pair of elements of F is separated by some hyperplane of H. We deal here with the case that the convex sets are all points. Let f(n, d) be the minimum number of hyperplanes always sufficient and occasionally necessary to separate n points in general position in Ed. We prove that [ (n −1) d ] ⩽ f(n, d) ⩽ [ (n −2 [ log (d)] ) d ] + [ log (d)] . When d is even the lower bound can be improved to [ n d ]. In the planar case this gives us f(n, 2) = [ n 2 ] . We prove the upper bound by presenting an algorithm that generates a separating family of hyperplanes H that satisfies the upper bound. In dimension 2 the algorithm has a time complexity of O(n log(n)). Finally, we show that in the planar case H can be stored such that retrieving a line in H that separates a given pair of points from P can be found in O(1) expected time and worst-case time of O(log2(n)).