Hyperplane Separability and Convexity of Probabilistic Point Sets

We describe an O(n) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d ≥ 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d + 1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [6]. In addition, our algorithms can handle “input degeneracies” in which more than k + 1 points may lie on a k-dimensional subspace, thus resolving an open problem in [6]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal. 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, F.2.2 Nonnumerical Algorithms and Problems, G.3 Probability and Statistics