Better Bounds on Online Unit Clustering.

Martin R. Ehmsen and Kim S. Larsen.
In 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), volume 6139 of Lecture Notes in Computer Science, pages 371-382. Springer, 2010.

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 from 2 to 13/6.

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