On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity
© 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....
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| Format: | Article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2021
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| Online Access: | https://hdl.handle.net/1721.1/137304 |
| _version_ | 1826207545761464320 |
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| author | Williams, Ryan |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Williams, Ryan |
| author_sort | Williams, Ryan |
| collection | MIT |
| description | © 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. |
| first_indexed | 2024-09-23T13:51:13Z |
| format | Article |
| id | mit-1721.1/137304 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:51:13Z |
| publishDate | 2021 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/1373042023-02-03T20:36:43Z On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity Williams, Ryan Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory © 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. 2021-11-03T18:42:38Z 2021-11-03T18:42:38Z 2018-01 2021-03-30T14:28:03Z Article http://purl.org/eprint/type/ConferencePaper https://hdl.handle.net/1721.1/137304 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. en 10.1137/1.9781611975031.78 Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms 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. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Williams, Ryan On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title | On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title_full | On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title_fullStr | On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title_full_unstemmed | On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title_short | On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity |
| title_sort | on the difference between closest furthest and orthogonal pairs nearly linear vs barely subquadratic complexity |
| url | https://hdl.handle.net/1721.1/137304 |
| work_keys_str_mv | AT williamsryan onthedifferencebetweenclosestfurthestandorthogonalpairsnearlylinearvsbarelysubquadraticcomplexity |