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

Sushmita Gupta, Shahin Kamali, Alejandro López-Ortiz

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 Θ(1)-competitive algorithms for a wide range of sparse graphs, which require advice of (almost) linear size. Namely, we show that for graphs of size N and treewidth α, there is an online algorithm which receives O(n(log α + log log N)) bits of advice and optimally serves a sequence of length n. With a different argument, we show that if a graph admits a system of μ collective tree (q,r)- spanners, then there is a (q + r)-competitive algorithm which receives O(n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, provided with O(n log log N) bits of advice. On the other side, we show that an advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of size n even for the 2-server problem on a path metric of size N ≥ 5. Through another lower bound argument, we show that at least \(\frac{n}{2}({\rm log} \alpha- 1.22)\) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 ≤ α < 2k.