Unbounded lower bound for k-server against weak adversaries

Bieńkowski, Marcin; Byrka, Jarosław; Coester, Christian; Jeż, Łukasz

doi:10.1145/3357713.3384306

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…In our problem, this extension could not just replace the restriction to the parameter m c , but also reduce the competitive ratio with respect to the number of servers. For Euclidean metrics, not much is known in this regard, with only a recent bound showing that no matter how high the difference in the number of servers, the dependence on the number of optimal servers can never be removed [6].…”

Section: Open Problemsmentioning

confidence: 99%

Managing Multiple Mobile Resources

et al. 2021

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We extend the Mobile Server problem introduced in Feldkord and Meyer auf der Heide (TOPC 6(3), 14:1–14:17 2019) to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. To reduce communication costs, the positions of the servers can be adapted by a limited distance ms per round for each server. The costs are measured similarly to the classical Page Migration problem: i.e., answering a request induces costs proportional to the distance to the nearest server, and moving a server induces costs proportional to the distance multiplied with a weight D. We show that, in our model, no online algorithm can have a constant competitive ratio: i.e., one which is independent of the input length n, even if an augmented moving distance of (1 + δ)ms is allowed for the online algorithm. Therefore we investigate a restriction of the power of the adversary dictating the sequence of requests: We demand locality of requests: i.e., that consecutive requests come from points in the Euclidean space with distance bounded by some constant mc. We show constant lower bounds on the competitiveness in this setting (independent of n, but dependent on k, ms and mc). On the positive side, we present a deterministic online algorithm with bounded competitiveness when an augmented moving distance and locality of requests is assumed. Our algorithm simulates any given algorithm for the classical k-Page Migration problem as guidance for its servers and extends it by a greedy move of one server in every round. The resulting competitive ratio is polynomial in the number of servers k, the ratio between mc and ms, the inverse of the augmentation factor 1/δ and the competitive ratio of the simulated k-Page Migration algorithm. We also show how to directly adapt the Double Coverage algorithm (Chrobak et al. SIAM J. Discrete Math. 4(2), 172–181 11) for the k-Server problem to receive an algorithm with improved competitiveness on the line.

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Section: Open Problemsmentioning

confidence: 99%

Managing Multiple Mobile Resources

et al. 2021

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“…Matching upper bounds appear in [17,24]. Previously, it has been widely conjectured (see [3,5,12,13,27,30,34,35,37,40,48]) that in all metric spaces on more than 𝑘 points, the randomized competitive ratio for the 𝑘server problem is Θ(log 𝑘). This was also established in some special cases [4,27,28].…”

Section: Introductionmentioning

confidence: 99%

The Randomized 𝑘-Server Conjecture Is False!

Bubeck

Coester

Rabani

2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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We prove a few new lower bounds on the randomized competitive ratio for the 𝑘-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987). More concretely, we show:(1) There exist (𝑘 + 1)-point metric spaces in which the randomized competitive ratio for the 𝑘-server problem is Ω(log 2 𝑘). This refutes the folklore conjecture (which is known to hold in some families of metrics) that in all metric spaces with at least 𝑘 + 1 points, the competitive ratio is Θ(log 𝑘).(2) Consequently, there exist 𝑛-point metric spaces in which the randomized competitive ratio for MTS is Ω(log 2 𝑛). This matches the upper bound that holds for all metrics. The previously best existential lower bound was Ω(log 𝑛) (which was known to be tight for some families of metrics). ( 3) For all 𝑘 < 𝑛 ∈ N, for all 𝑛-point metric spaces the randomized 𝑘-server competitive ratio is at least Ω(log 𝑘), and consequently the randomized MTS competitive ratio is at least Ω(log 𝑛). These universal lower bounds are asymptotically tight. The previous bounds were Ω(log 𝑘/log log 𝑘) and Ω(log 𝑛/log log 𝑛), respectively. (4) The randomized competitive ratio for the 𝑤-set metrical service systems problem, and its equivalent width-𝑤 layered graph traversal problem, is Ω(𝑤 2 ). This slightly improves the previous lower bound and matches the recently discovered upper bound. (5) Our results imply improved lower bounds for other problems like 𝑘-taxi, distributed paging, and metric allocation.These lower bounds share a common thread, and other than the third bound, also a common construction.

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The Infinite Server Problem

Coester

Koutsoupias

Lazos

2021

ACM Trans. Algorithms

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We study a variant of the k -server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k -server problem, also known as the (h,k) -server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k) -server problem has bounded competitive ratio for some k = O ( h ). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k) -server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

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Unbounded lower bound for k-server against weak adversaries

Cited by 3 publications

References 21 publications

Managing Multiple Mobile Resources

Managing Multiple Mobile Resources

The Randomized 𝑘-Server Conjecture Is False!

The Infinite Server Problem

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