Tight Bounds for Double Coverage Against Weak Adversaries

We study the Double Coverage (DC) algorithm for the k -server problem in tree metrics in the ( h , k )-setting, i.e., when DC with k servers is compared against an offline optimum algorithm with h ≤ k servers. It is well-known that in such metric spaces DC is k -competitive (and thus optimal) for h = k . We prove that even if k > h the competitive ratio of DC does not improve; in fact, it increases slightly as k grows, tending to h + 1. Specifically, we give matching upper and lower bounds of k(h+1)/k+1 on the competitive ratio of DC on any tree metric.