On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity
Author(s)
Williams, Ryan
DownloadPublished version (385.2Kb)
Publisher Policy
Publisher Policy
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Terms of use
Metadata
Show full item recordAbstract
© Copyright 2018 by SIAM. Point location problems for n points in d-dimensional Euclidean space (and 'p spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than dpoly(d) n logO(d) n time, or (Barely-Subquadratic) f(d)n2-1= (d) time, for various f. For small d and large n, \nearly-linear" running times are generally feasible, while the \barely-subquadratic" times are generally infeasible, requiring essentially quadratic time. For example, in the Euclidean metric, finding a Closest Pair among n points in Rd is nearly-linear, solvable in 2O(d) n logO(1) n time, while the known algorithms for finding a Furthest Pair (the diameter of the point set) are only barelysubquadratic, requiring (n2-1=(d)) time. Why do these proximity problems have such different time complexities? Is there a barrier to obtaining nearly-linear algorithms for problems which are currently only barely-subquadratic? We give a novel exact and deterministic self-reduction for the Orthogonal Vectors problem on n vectors in f0; 1gd to n vectors in Z!(log d) that runs in 2o(d) time. As a consequence, barely-subquadratic problems such as Euclidean diameter, Euclidean bichromatic closest pair, and incidence detection do not have O(n2-1) time algorithms (in Turing models of computation) for dimensionality d = (log log n)2, unless the popular Orthogonal Vectors Conjecture and the Strong Exponential Time Hypothesis are false. That is, while the poly-log-log-dimensional case of Closest Pair is solvable in n1+o(1) time, the poly-log-log-dimensional case of Furthest Pair can encode difficult large-dimensional problems conjectured to require n2-o(1) time. We also show that the All-Nearest Neighbors problem in !(log n) dimensions requires n2-o(1) time to solve, assuming either of the above conjectures.
Date issued
2018-01Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence LaboratoryJournal
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Williams, Ryan. 2018. "On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity." Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms.
Version: Final published version