On the Online Unit Clustering Problem

We continue the study of the online unit clustering problem, introduced by Chan and Zarrabi-Zadeh (\emph{Workshop on Approximation and Online Algorithms 2006}, LNCS 4368, p.121--131. Springer, 2006). We design a deterministic algorithm with a competitive ratio of $7/4$ for the one-dimensional case. This is the first deterministic algorithm that beats the bound of 2. It also has a better competitive ratio than the previous randomized algorithms. Moreover, we provide the first non-trivial deterministic lower bound, improve the randomized lower bound, and prove the first lower bounds for higher dimensions