On the Separability of Stochastic Geometric Objects, with Applications

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \cup S_B$ be a given set of stochastic bichromatic points, and define $n = \min\{|S_R|, |S_B|\}$ and $N = \max\{|S_R|, |S_B|\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \geq 3$ and $O(\min\{nN \log N, N^2\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \geq 2$. In addition, we give an $\Omega(nN^{d-1})$ witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in $O(nN^{d})$ and $O(nN^{d+1})$ time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.