On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest

Algorithms and hardness results for geometric problems on stochastic datasets

University of Minnesota Ph.D. dissertation.July 2019. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); viii, 121 pages.Traditionally, geometric problems are studied on datasets in which each data object exists with probability 1 at its location in the underlying space. However, in many scenarios, there may be some uncertainty associated with the existence or the locations of the data points. Such uncertain datasets, called \textit{stochastic datasets}, are often more realistic, as they are more expressive and can model the real data more precisely. For this reason, geometric problems on stochastic datasets have received significant attention in recent years. This thesis studies three sets of geometric problems on stochastic datasets equipped with existential uncertainty. The first set of problems addresses the linear separability of a bichromatic stochastic dataset. Specifically, these problems are concerned with how to compute the probability that a realization of a bichromatic stochastic dataset is linearly separable as well as how to compute the expected separation-margin of such a realization. The second set of problems deals with the stochastic convex hull, i.e., the convex hull of a stochastic dataset. This includes computing the expected measures of a stochastic convex hull, such as the expected diameter, width, and combinatorial complexity. The third set of problems considers the dominance relation in a colored stochastic dataset. These problems involve computing the probability that a realization of a colored stochastic dataset does not contain any dominance pair consisting of two different-colored points. New algorithmic and hardness results are provided for the three sets of problems