New bounds for randomized list update in the paid exchange model

We study the fundamental list update problem in the paid exchange model P^d. This cost model was introduced by Manasse, McGeoch and Sleator [18] and Reingold, Westbrook and Sleator [24]. Here the given list of items may only be rearranged using paid exchanges; each swap of two adjacent items in the list incurs a cost of d. Free exchanges of items are not allowed. The model is motivated by the fact that, when executing search operations on a data structure, key comparisons are less expensive than item swaps. We develop a new randomized online algorithm that achieves an improved competitive ratio against oblivious adversaries. For large d, the competitiveness tends to 2.2442. Technically, the analysis of the algorithm relies on a new approach of partitioning request sequences and charging expected cost. Furthermore, we devise lower bounds on the competitiveness of randomized algorithms against oblivious adversaries. No such lower bounds were known before. Specifically, we prove that no randomized online algorithm can achieve a competitive ratio smaller than 2 in the partial cost model, where an access to the i-th item in the current list incurs a cost of i − 1 rather than i. All algorithms proposed in the literature attain their competitiveness in the partial cost model. Furthermore, we show that no randomized online algorithm can achieve a competitive ratio smaller than 1.8654 in the standard full cost model. Again the lower bounds hold for large d.