High-Dimensional Approximate r-Nets

The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r -nets with respect to Euclidean distance. For any fixed ϵ > 0 , the approximation factor is...

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Bibliographic Details
Published in:Algorithmica Vol. 82; no. 6; pp. 1675 - 1702
Main Authors: Avarikioti, Z., Emiris, I. Z., Kavouras, L., Psarros, I.
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2020
Springer Nature B.V
Springer Verlag
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Clustering
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Euclidean geometry
Geometry
Mathematics
Mathematics of Computing
Metric Geometry
Metric space
Polynomials
Software
Theory of Computation
General dimension
Locality sensitive hashing
Approximation algorithm
nets
Metric geometry
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r -nets with respect to Euclidean distance. For any fixed ϵ > 0 , the approximation factor is 1 + ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555 ) in terms of complexity, by reducing the dependence on ϵ , provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167 ) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r -nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the ( 1 + ϵ ) -approximate k -th nearest neighbor distance in time subquadratic in the size of the input.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00664-8