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

Abstract

AbstractWe 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.KeywordsCompetitive RatioOnline AlgorithmTree DecompositionChordal GraphUnit Disk GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.