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 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, under a reasonable streaming model, 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, with high probability, an -hull for all but δ-fraction of directions and requires O(OPT · log OPT) space. 1 This work was supported in part by the National Science Foundation under grant CCF-1525971. Work was done while the author was at Carnegie Mellon University. 2 This material is based upon work supported in part by the National Science Foundation under Grants No. 1447639, 1650041 and 1652257, Cisco faculty award, and by the ONR Award N00014-18-1-2364. 3 Now at DeepMind. 4 This research was supported by the Franco-American Fulbright Commission and supported in part by National Science Foundation under Grant No. 1447639, 1650041 and 1652257. The author thanks INRIA (l’Institut national de recherche en informatique et en automatique) for hosting him during the writing of this paper. 5 This material is based upon work supported in part by National Science Foundation under Grant No. 1447639, 1650041 and 1652257. Work was done while the author was at Johns Hopkins University. EA T C S © Avrim Blum, Vladimir Braverman, Ananya Kumar, Harry Lang, and Lin F. Yang; licensed under Creative Commons License CC-BY 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella; Article No. 21; pp. 21:1–21:13 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 21:2 Approximate Convex Hull of Data Streams 2012 ACM Subject Classification Theory of computation → Computational geometry, Theory of computation → Sketching and sampling, Theory of computation → Streaming models