Online Scheduling for Sorting Buffers

We introduce the online scheduling problem for sorting buffers. A service station and a sorting buffer are given. An input sequence of items which are only characterized by a specific attribute has to be processed by the service station which benefits from consecutive items with the same attribute value. The sorting buffer which is a random access buffer with storage capacity for $k$ items can be used to rearrange the input sequence. The goal is to minimize the cost of the service station, i.e., the number of maximal subsequences in its sequence of items containing only items with the same attribute value. This problem is motivated by many applications in computer science and economics.

The strategies are evaluated in a competitive analysis in which the cost of the online strategy is compared with the cost of an optimal offline strategy. Our main result is a deterministic strategy that achieves a competitive ratio of $O\left({log}^{2}k\right)$. In addition, we show that several standard strategies are unsuitable for this problem, i.e., we prove a lower bound of $\Omega \left(\sqrt{k}\right)$ on the competitive ratio of the First In First Out (FIFO) and Least Recently Used (LRU) strategy and of $\Omega \left(k\right)$ on the competitive ratio of the Largest Color First (LCF) strategy.