The weighted list update problem and the lazy adversary

Abstract

The list update problem consists in maintaining a dictionary as an unsorted linear list. Any request specifies an item to be found by sequential scanning through the list. After an item has been found, the list may be rearranged in order to reduce the cost of processing a sequence of requests. Several kinds of adversaries can be considered to analyze the behavior of heuristics for this problem. The move-to-front (MTF) heuristic is 2-competitive against a strong adversary, matching the deterministic lower bound for this problem [Sleator and Tarjan (1985)]. But, for this problem, moving elements does not help the adversary. A lazy adversary has the limitation that he can use only a static arrangement of the list to process (off-line) the sequence of requests: still, no algorithm can be better than 2-competitive against the lazy adversary [Bentley and McGeogh (1985)]. In this paper we consider the weighted list update problem (WLUP), where the cost of accessing an item depends on the item itself. It is shown that MTF is not competitive by any constant factor for this problem against a lazy adversary. Two heuristics, based on the MTF strategy, are presented for WLUP: random move-to-front is randomized and uses biased coins; counting move-to-front is deterministic, and replaces coins by counters. Both are shown to be 2-competitive against a lazy adversary. This is optimal for the deterministic case. We apply this approach for searching items in a tree, proving that any c-competitive heuristic for the weighted list update problem provides a c-competitive heuristic for the tree update problem.