We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of @math points in @math 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 @math and @math . For width, two approximation algorithms are provided: a deterministic @math -approximation running in @math time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact @math -time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.
Geometric computation on uncertain data has received considerable attentions in recent years. A general introduction can be found in @cite_12 . Many fundamental geometric problems have been studied under uncertainty, e.g., nearest-neighbor search @cite_4 @cite_14 , minimum spanning trees @cite_9 , closest pair @cite_17 @cite_7 @cite_10 , range search @cite_18 @cite_15 , linear separability @cite_2 @cite_16 , dominance relation @cite_8 , etc. These problems were studied either under existential uncertainty (which is used in this paper) or under locational uncertainty (where the locations of the data points are uncertain).
@cite_18 @cite_14 @cite_4 @cite_7 @cite_8 @cite_9 @cite_2 @cite_15 @cite_16 @cite_10 @cite_12 @cite_17
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We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, (k )-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be ( # P )-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge. With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an epsilon-EXP-KERNEL), as well as the probability distribution on the width (an (epsilon, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(epsilon^ -(d-1) 2 ) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an epsilon-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (epsilon, tau)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty. Uncertain data has been very common in many applications. In this paper, we consider the one-center problem for uncertain data on tree networks. In this problem, we are given a tree T and n (weighted) uncertain points each of which has m possible locations on T associated with probabilities. The goal is to find a point @math xź on T such that the maximum (weighted) expected distance from @math xź to all uncertain points is minimized. To the best of our knowledge, this problem has not been studied before. We propose a refined prune-and-search technique that solves the problem in linear time. We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.