- Better Bounds on Online Unit Clustering.
- Martin R. Ehmsen and Kim S. Larsen.
Theoretical Computer Science, 500:1-24, 2013.
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 the work, recently 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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