The (h, k)-server problem on bounded depth trees

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, that is believed to be optimal for all metric spaces when k = h, has competitive ratio Ω(h) on depth-3 HSTs even if k = 2h. 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, and gives the strongest known lower bound for the problem on general metrics.