Abstract
Unit Clustering is the problem of dividing a set of points from a metric space into a minimal number of subsets such that the points in each subset are enclosable by a unit ball. We continue work initiated by Chan and Zarrabi-Zadeh on determining the competitive ratio of the online version of this problem. For the one-dimensional case, we develop a deterministic algorithm, improving the best known upper bound of 7/4 by Epstein and van Stee to 5/3. This narrows the gap to the best known lower bound of 8/5 to only 1/15. Our algorithm automatically leads to improvements in all higher dimensions as well. Finally, we strengthen the deterministic lower bound in two dimensions and higher fromĀ 2 to 13/6.
This work was supported in part by the Danish Natural Science Research Council.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theory of Computing SystemsĀ 45(3), 486ā496 (2009)
Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM Journal on ComputingĀ 33(6), 1417ā1440 (2004)
Ehmsen, M.R., Larsen, K.S.: Better Bounds on Online Unit Clustering. PreprintĀ 8, Department of Mathematics and Computer Science, University of Southern Denmark (2009)
Epstein, L., Levin, A., van Stee, R.: Online unit clustering: Variations on a theme. Theoretical Computer ScienceĀ 407, 85ā96 (2008)
Epstein, L., van Stee, R.: On the online unit clustering problem. In: Proceedings of the 5th International Workshop on Approximation and Online Algorithms, pp. 193ā206 (2007)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Systems Technical JournalĀ 45, 1563ā1581 (1966)
Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. AlgorithmicaĀ 3, 79ā119 (1988)
Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACMĀ 28(2), 202ā208 (1985)
Zarrabi-Zadeh, H., Chan, T.M.: An improved algorithm for online unit clustering. AlgorithmicaĀ 54(4), 490ā500 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ehmsen, M.R., Larsen, K.S. (2010). Better Bounds on Online Unit Clustering. In: Kaplan, H. (eds) Algorithm Theory - SWAT 2010. SWAT 2010. Lecture Notes in Computer Science, vol 6139. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13731-0_35
Download citation
DOI: https://doi.org/10.1007/978-3-642-13731-0_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13730-3
Online ISBN: 978-3-642-13731-0
eBook Packages: Computer ScienceComputer Science (R0)