On the randomized competitive ratio of reordering buffer management with non-uniform costs

Reordering buffer management (RBM) is an elegant theoretical model that captures the tradeoff between buffer size and switching costs for a variety of reordering/sequencing problems. In this problem, colored items arrive over time, and are placed in a buffer of size k. When the buffer becomes full, an item must be removed from the buffer. A penalty cost is incurred each time the sequence of removed items switches colors. In the non-uniform cost model, there is a weight wc associated with each color c, and the cost of switching to color c is wc. The goal is to minimize the total cost of the output sequence, using the buffer to rearrange the input sequence. Recently, a randomized O(log log k)-competitive online algorithm was given for the case that all colors have the same weight (FOCS 2013). This is an exponential improvement over the nearly tight bound of O(√ log k) on the deterministic competitive ratio of that version of the problem (Adamaszek et al., STOC 2011). In this paper, we give an O((log log kγ)²)-competitive algorithm for the non-uniform case, where γ is the ratio of the maximum to minimum color weight. Our work demonstrates that randomness can achieve exponential improvement in the competitive ratio even for the non-uniform case.