A Polylogarithmic-Competitive Algorithm for the
            <i>k</i>
            -Server Problem

Bansal, Nikhil; Buchbinder, Niv; Mądry, Aleksander; Naor, Joseph

doi:10.1145/2783434

Cited by 76 publications

(93 citation statements)

References 25 publications

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“…In the area of online algorithms, the method of modeling a problem as a linear program, obtaining a fractional solution via a primal-dual algorithm, and then rounding it, has proved to be a very powerful general technique (see [4] for a survey). This framework is applicable to many central online problems, like the classic ski rental problem, online set-cover [5], generalized paging [6], [7], k-server and metrical task systems [8], [9], graph connectivity [10], [11], routing [12], load balancing and machine scheduling [13], [14], matching [15], and budgeted allocation [16]. Via this approach, not only can we unify previously known results, but we can also resolve important open questions in competitive analysis.…”

Section: Introductionmentioning

confidence: 99%

Online Algorithms for Covering and Packing Problems with Convex Objectives

Azar

Buchbinder

Chan

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min x∈R n + f (x) s.t. Ax ≥ 1, where f : R n + → R+ is a monotone convex function, and A is an m × n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: max y∈R m + m j=1 yj − g(A T y), where g : R n + → R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

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Section: Introductionmentioning

confidence: 99%

Online Algorithms for Covering and Packing Problems with Convex Objectives

Azar

Buchbinder

Chan

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

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“…It is speculated that this factor can be obtained for all metrics, however the question is still open. For general metrics, the first algorithm with polylogarithmic competitive ratio was an O(log 3 n • log 2 k)-competitive algorithm by Bansal et al [3]. This was recently improved by Bubeck et al [10] who gave an O(log 2 k)-competitive algorithm for HSTs which can be turned into an O(log 9 (k) • log log(k))-competitive one for general metrics by a dynamic embedding of general metrics into HSTs [17].…”

Section: Related Workmentioning

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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“…There is a vast literature on the k-server and RBM problems, see for example [1], [12], [16], [29], [31] and [4], [13], [19], [20], [22], [24], [25], [26], [27], [28] respectively. We limit our discussion to the results most closely related to ours.…”

Section: Further Related Workmentioning

confidence: 99%

Polylogarithmic Guarantees for Generalized Reordering Buffer Management

Englert

Räcke

Stotz

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In the Generalized Reordering Buffer Management Problem (GRBM) a sequence of items located in a metric space arrives online, and has to be processed by a set of k servers moving within the space. In a single step the first b still unprocessed items from the sequence are accessible, and a scheduling strategy has to select an item and a server. Then the chosen item is processed by moving the chosen server to its location. The goal is to process all items while minimizing the total distance travelled by the servers.This problem was introduced in [Chan, Megow, Sitters, van Stee TCS 12] and has been subsequently studied in an online setting by [Azar, Englert, Gamzu, Kidron STACS 14]. The problem is a natural generalization of two very well-studied problems: the k-server problem for b = 1 and the Reordering Buffer Management Problem (RBM) for k = 1. In this paper we consider the GRBM problem on a uniform metric in the online version. We show how to obtain a competitive ratio of O(log k(log k + log log b)) for this problem. Our result is a drastic improvement in the dependency on b compared to the previous best bound of O( √ b log k), and is asymptotically optimal for constant k, because Ω(log k+log log b) is a lower bound for GRBM on uniform metrics.

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A Polylogarithmic-Competitive Algorithm for the k -Server Problem

Cited by 76 publications

References 25 publications

Online Algorithms for Covering and Packing Problems with Convex Objectives

Online Algorithms for Covering and Packing Problems with Convex Objectives

Managing Multiple Mobile Resources

Polylogarithmic Guarantees for Generalized Reordering Buffer Management

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