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

We consider the k -S erver problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce Θ(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 α , there is an online algorithm that receives O ( n (log α + 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 ≥ 3. Through another lower bound argument, we show that at least n/2(logα - 1.22) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α , where 4 ≤ α < 2 k .