On the online unit clustering problem

We continue the study of the online unit clustering problem, introduced by Chan and Zarrabi-Zadeh (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.