Abstract
Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are k-Means, Geometric k-Median/Center, Continuous Facility Location, m-Variance, etc. We prove that, for any fixed \(\varepsilon >0\), every set of n input points in fixed-dimensional space with the metric induced by any vector norm admits a set of O(n) candidate centers which can be computed in almost linear time and which contains a \((1+\varepsilon )\)-approximation of each point of space with respect to the distances to all the input points. It gives a universal approximation-preserving reduction of geometric center-based problems with arbitrary continuity-type objective functions to their discrete versions where the centers are selected from a fairly small set of candidates. The existence of such a linear-size set of candidates is also shown for any metric space of fixed doubling dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal P, Procopiuc C (2002) Exact and approximation algorithms for clustering. Algorithmica 33(2):201–226. https://doi.org/10.1007/s00453-001-0110-y
Agarwal P, Har-Peled S, Varadarajan K (2005) Geometric approximation via coresets. In: Goodman JE, Pach J, Welzl E (eds) Combinatorial and computational geometry, vol 52. Cambridge University Press, Cambridge, pp 1–30. http://library.msri.org/books/Book52/files/01agar.pdf
Aggarwal A, Imai H, Katoh N, Suri S (1991) Finding \(k\) points with minimum diameter and related problems. J Algorithms 12(1):38–56. https://doi.org/10.1016/0196-6774(91)90022-Q
Alon N, Lee T, Shraibman A, Vempala S (2013a) The approximate rank of a matrix and its algorithmic applications. Extended version of the STOC 2013 paper. https://www.cc.gatech.edu/~vempala/papers/epsrank.pdf
Alon N, Lee T, Shraibman A, Vempala S (2013b) The approximate rank of a matrix and its algorithmic applications. In: Proceedings of 45th ACM symposium on theory of computing (STOC 2013), pp 675–684. https://doi.org/10.1145/2488608.2488694
Arora S, Raghavan P, Rao S (1998) Approximation schemes for Euclidean \(k\)-medians and related problems. In: Proceedings of 30th ACM symposium on theory of computing (STOC 1998), pp 106–113. https://doi.org/10.1145/276698.276718
Bǎdoiu M, Har-Peled S, Indyk P (2002) Approximate clustering via core-sets. In: Proceedings of 34th ACM symposium on theory of computing (STOC 2002), pp 250–257. https://doi.org/10.1145/509907.509947
Callahan P, Kosaraju S (1995) A decomposition of multidimensional point sets with applications to \(k\)-nearest-neighbors and \(n\)-body potential fields. J ACM 42(1):67–90. https://doi.org/10.1145/200836.200853
Chen K (2006) On \(k\)-median clustering in high dimensions. In: Proceedings of 17th ACM-SIAM symposium on discrete algorithms (SODA 2006), pp 1177–1185. https://doi.org/10.5555/1109557.1109687
Eppstein D, Erickson J (1994) Iterated nearest neighbors and finding minimal polytopes. Disc Comput Geom 11(3):321–350. https://doi.org/10.1007/BF02574012
Har-Peled S, Mendel M (2006) Fast construction of nets in low-dimensional metrics and their applications. SIAM J Comput 35(5):1148–1184. https://doi.org/10.1137/S0097539704446281
Har-Peled S, Mazumdar S (2004) On coresets for \(k\)-means and \(k\)-median clustering. In: Proceedings 36th ACM symposium on theory of computing (STOC 2004), pp 291–300. https://doi.org/10.1145/1007352.1007400
Jaiswal R, Kumar A, Sen S (2014) A simple \(D^2\)-sampling based PTAS for \(k\)-means and other clustering problems. Algorithmica 70(1):22–46. https://doi.org/10.1007/s00453-013-9833-9
Kelmanov A, Motkova A, Shenmaier V (2018) Approximation scheme for the problem of weighted \(2\)-clustering with a fixed center of one cluster. Proc Steklov Inst Math 303(1):136–145. https://doi.org/10.1134/S0081543818090146
Kumar P, Mitchell J, Yıldırım E (2003) Approximate minimum enclosing balls in high dimensions using core-sets. J Exp Algorithmics 8:1–29. https://doi.org/10.1145/996546.996548
Kumar A, Sabharwal Y, Sen S (2010) Linear-time approximation schemes for clustering problems in any dimensions. J ACM 57(2):1–32. https://doi.org/10.1145/1667053.1667054
Matoušek J (2000) On approximate geometric \(k\)-clustering. Discrete Comput Geom 24(1):61–84. https://doi.org/10.1007/s004540010019
Meira A, Miyazawa F, Pedrosa L (2017) Clustering through continuous facility location problems. Theor Comput Sci 657(B):137–145. https://doi.org/10.1016/j.tcs.2016.10.001
Shenmaier V (2019) A structural theorem for center-based clustering in high-dimensional Euclidean space. In: Proceedings of 5th conference on machine learning, optimization, and data science (LOD 2019), LNCS 11943, pp 284–295. https://doi.org/10.1007/978-3-030-37599-7_24
Shenmaier V (2020) Some estimates on the discretization of geometric center-based problems in high dimensions. In: Proceedings of 19th conference mathematical optimization theory and operations research (MOTOR 2020). CCIS 127, pp 88–101. https://doi.org/10.1007/978-3-030-58657-7_10
Shenmaier V (2012) An approximation scheme for a problem of search for a vector subset. J Appl Industr Math 6(3):381–386. https://doi.org/10.1134/S1990478912030131
Shenmaier V (2015) Complexity and approximation of the smallest \(k\)-enclosing ball problem. Eur J Comb 48:81–87. https://doi.org/10.1016/j.ejc.2015.02.011
Smid M (2007) The well-separated pair decomposition and its applications. In: Gonzalez TF (ed) Handbook of approximation algorithms and metaheuristics. Taylor & Francis, New York, pp 53.1–53.12. https://doi.org/10.1201/9781420010749-63
Talwar K (2004) Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of 36th ACM symposium on theory of computing (STOC 2004), pp 281–290. https://doi.org/10.1145/1007352.1007399
Acknowledgements
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (Project 0314-2019-0014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shenmaier, V. Linear-size universal discretization of geometric center-based problems in fixed dimensions. J Comb Optim 43, 528–542 (2022). https://doi.org/10.1007/s10878-021-00790-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-021-00790-6