The online unit clustering problem is one of the most basic clustering problems proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009). Several variants of this problem have been extensively studied. In this letter, we propose a new variant of the online unit clustering problem, called the online unit clustering problem with capacity constraints. For this problem, we use competitive analysis to evaluate the performance of an online algorithm. Then, we develop an online algorithm whose competitive ratio is at most 3.178, and show that a lower bound on the competitive ratio of any online algorithm is 2.

The online interval coloring problem has been extensively studied for many years. Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for this problem. The number of colors used by the algorithm is at most 3ω(G)-2, where ω(G) is the size of the maximum clique in a given graph G. Also, they presented an instance for which the number of colors used by any online algorithm is at least 3ω(G)-2. This instance includes intervals with various lengths, which cannot be applied to the case when the lengths of the given intervals are restricted to one, i.e., the online unit interval coloring problem. In this case, the current best upper and lower bounds on the number of colors used by an online algorithm are 2ω(G)-1 and 3ω(G)/2 respectively by Epstein and Levy (ICALP2005). In this letter, we conduct a complete performance analysis of the Kierstead-Trotter algorithm for online unit interval coloring, and prove it is NOT optimal. Specifically, we provide an upper bound of 3ω(G)-3 on the number of colors used by their algorithm. Moreover, the bound is the best possible.