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 (ℎ+1)/(+1) on the competitive ratio of DC on any tree metric.