On the Advice Complexity of the k-server Problem Under Sparse Metrics

Abstract

We consider the k-Server problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce ź(1)-competitive algorithms for a wide range of sparse graphs. These algorithms require advice of (almost) linear size. We show that for graphs of size N and treewidth ź, there is an online algorithm that receives O (n(log ź + log log N))* bits of advice and optimally serves any sequence of length n. We also prove that if a graph admits a system of μ collective tree (q, r)-spanners, then there is a (q + r)-competitive algorithm which requires O (n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, when provided with O (n log log N) bits of advice. On the other side, we prove that advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of length n even for the 2-server problem on a path metric of size N ź 3. Through another lower bound argument, we show that at least n2(logźź1.22)$\frac {n}{2}(\log \alpha - 1.22)$ bits of advice is required to obtain an optimal solution for metric spaces of treewidth ź, where 4 ≤ ź < 2k.