We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

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.