Work function algorithm can forget history without losing competitiveness

The Work Function Algorithm (WFA) is the most effective deterministic on-line algorithm for the k-server problem. E. Koutsoupias and C. Papadimitriou in [6] proved that WFA is (2k − 1)-competitive and it is conjectured that it is k-competitive. However the best known implementation of WFA requires time O(i2) to process request ri and this makes WFA impractical for long sequences of requests. The O(i2) time is spent to compute the work function on the whole history r1, . . . , ri of past requests. In order to make constant the time to process a request, Rudec and Menger in [7] proposed to restrict the history to a moving window of fixed size. However WFA restricted to a moving window loses its competitiveness [9]. Here we give a condition that allows WFA to forget the whole previous history and restart from scratch without losing competitiveness. Moreover for most of the metric spaces of practical interest (finite or bounded spaces) there is a constant bound on the length of the history before the condition is verified and this makes O(1) the time to process each request. The condition is first given for on-line algorithms in the more general framework of Metrical Task Systems (MTS) [4, 5] ad then it is restricted to the k-server problem.