$ε$-Kernel Coresets for Stochastic Points

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 ε-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 εkernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an ε-exp-kernel), as well as the probability distribution on the width (an (ε, τ)-quant-kernel) for any direction. We show that there exists a set of O(ε−(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 ε-exp-kernel is still possible. We also provide efficient algorithms for construct an (ε, τ)-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. 1998 ACM Subject Classification B.2.4 Algorithms, F.2.2 Geometrical problems and computations