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

• Published in: Theory of computing systems Vol. 59; no. 3; pp. 476 - 499
• Main Authors: Gupta, Sushmita, Kamali, Shahin, López-Ortiz, Alejandro
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2016; Springer Nature B.V
• Subjects: Algorithms; Competition; Computation; Computational mathematics; Computer Science; Graph coloring; Graphs; Mathematical models; Metric space; Optimization; Servers; Studies; Texts; Theory of Computation; Trees; Advice complexity; Competitive analysis; Server problem
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-015-9649-x

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
(
log
α
−
1.22
)
bits of advice is required to obtain an optimal solution for metric spaces of treewidth
α
, where 4 ≤
α
< 2
k
.