Approximate Convex Hull of Data Streams

Given a finite set of points P ⊆ R, we would like to find a small subset S ⊆ P such that the convex hull of S approximately contains P . More formally, every point in P is within distance from the convex hull of S. Such a subset S is called an -hull. Computing an -hull is an important problem in computational geometry, machine learning, and approximation algorithms. In many real world applications, the set P is too large to fit in memory. We consider the streaming model where the algorithm receives the points of P sequentially and strives to use a minimal amount of memory. Existing streaming algorithms for computing an -hull require O( (1−d)/2) space, which is optimal for a worst-case input. However, this ignores the structure of the data. The minimal size of an -hull of P , which we denote by OPT, can be much smaller. A natural question is whether a streaming algorithm can compute an -hull using only O(OPT) space. We begin with lower bounds that show that it is not possible to have a single-pass streaming algorithm that computes an -hull with O(OPT) space. We instead propose three relaxations of the problem for which we can compute -hulls using space near-linear to the optimal size. Our first algorithm for points in R2 that arrive in random-order uses O(logn · OPT) space. Our second algorithm for points in R2 makes O(log( 1 )) passes before outputting the -hull and requires O(OPT) space. Our third algorithm for points in R for any fixed dimension d outputs an -hull for all but δ-fraction of directions and requires O(OPT · log OPT) space. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems