The server problem and on-line games

Chrobák, Marek; Larmore, Lawrence L.

doi:10.1090/dimacs/007/02

Cited by 73 publications

(77 citation statements)

References 19 publications

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“…The right part is deterministic, and accepts words of the form w# r w, for w ∈ {a, b} r , at a cost of r. The left part is nondeterministic and accepts words of the form w# r v, for w, v ∈ {a, b} r , at a cost of α · r. After its nondeterministic branches, the left part has 2 r branches, such that for every word v ∈ {a, b} r there is a distinct branch that accepts v at a cost of α · r and accepts all other words in {a, b} r at a cost greater than α · r but at most α 2 · r. This is achieved by generalizing the costs 4 and 6 in A 4,3 by the costs α and β, respectively, for α < β ≤ α 2 . Theorem 5.2 also follows from specific examples studied in the literature, showing that online algorithms that can store additional information can achieve better competitive ratios (for example, [6] shows a lower bound of 23/11 on the competitiveness of any deterministic trackless online algorithm for the 2-server problem 5 ; whereas [10] shows that the competitive ratio of the Work Function Algorithm, which is also deterministic, but not trackless, for the 2-server problem is 2). Nonetheless, the proof of Theorem 5.2 serves to pinpoint the source of this phenomenon.…”

Section: Proofmentioning

confidence: 83%

Reasoning about online algorithms with weighted automata

Aminof

Kupferman

Lampert

2010

ACM Trans. Algorithms

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We describe an automata-theoretic approach for the competitive analysis of online algorithms. Our approach is based on weighted automata, which assign to each input word a cost in IR ≥0 . By relating the "unbounded look ahead" of optimal offline algorithms with nondeterminism, and relating the "no look ahead" of online algorithms with determinism, we are able to solve problems about the competitive ratio of online algorithms, and the memory they require, by reducing them to questions about determinization and approximated determinization of weighted automata.

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Section: Proofmentioning

confidence: 83%

Reasoning about online algorithms with weighted automata

Aminof

Kupferman

Lampert

2010

ACM Trans. Algorithms

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“…To this end we make use of a state diagram, called the offset graph, which indicates the value of the work function W (s, σ) [25]. Recall that W (s, σ) gives the optimal offline cost under the assumption that all requests given by σ have been served and the final state after σ must be s, where s is one of (L, C), (L, R) and (C, R) in our situation.…”

Section: Three Points On a Linementioning

confidence: 99%

Randomized Competitive Analysis for Two Server Problems

2008

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“…Given a sequence of tasks σ we define the work function [13] at v, w σ,U (v), to be the minimal cost, for any off-line player, to start at the initial state in U , deal with all tasks in σ, and end up in state v. We omit the use of the subscript U if it is clear from the context. Note that for all u, v ∈ M , w σ (u) − w σ (v) ≤ d M (u, v).…”

Section: Preliminariesmentioning

confidence: 99%

Better Algorithms for Unfair Metrical Task Systems and Applications

Fiat¹,

Mendel²

2003

SIAM J. Comput.

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Unfair metrical task systems are a generalization of online metrical task systems. In this paper we introduce new techniques to combine algorithms for unfair metrical task systems and apply these techniques to obtain improved randomized online algorithms for metrical task systems on arbitrary metric spaces. IntroductionMetrical task systems, introduced by Borodin, Linial, and Saks [11], can be described as follows: A server in some internal state receives tasks that have a service cost associated with each of the internal states. The server may switch states, paying a cost given by a metric space defined on the state space, and then pays the service cost associated with the new state.Metrical task systems have been the subject of a great deal of study. A large part of the research into online algorithms can be viewed as a study of some particular metrical task system. In modelling some of these problems as metrical task systems, the set of permissible tasks is constrained to fit the particulars of the problem. In this paper we consider the original definition of metrical task systems where the set of tasks can be arbitrary. A deterministic algorithm for any n-state metrical task system with a competitive ratio of 2n−1 is given in [11], along with a matching lower bound for any metric space.The randomized competitive ratio of the MTS problem is not as well understood. For the uniform metric space, where all distances are equal, the randomized competitive ratio is known to within a constant factor, and is Θ(log n) [11,14]. In fact, it has been conjectured that the randomized competitive ratio for MTS is Θ(log n) in any n-point metric space. Previously, the best upper bound on the competitive ratio for arbitrary n-point metric space was O(log 5 n log log n) due Bartal, Blum, Burch and Tomkins [3] and Bartal [2]. The best lower for any n-point metric space is Ω(log n/ log log n) due to Bartal, Bollobás and Mendel [4] and Bartal, Linial, Mendel and Naor [5], improving previous lower bounds of Karloff, Rabani and Ravid [16], and Blum, Karloff, Rabani, and Saks [10].As observed in [16,10,1] , we obtain an improved algorithm for HSTs. In order to reduce the MTS problem on arbitrary metric space to a MTS problem on a HST we use probabilistic embedding of metric spaces into HSTs [1]. It is shown in [2] that any n-point metric space has probabilistic embedding in k-HSTs with distortion O(k log n log log n). Thus, an MTS problem on an arbitrary n-point metric space, can be reduced to an MTS problem on a k-HST with overhead of O(k log n log log n) [1].Our algorithm for HSTs follows the general framework given in [10] and explicitly formulated in [18,3], where the recursive structure of the HST is modelled by defining an unfair metrical task system problem [18, 3] on a uniform metric space. In an unfair MTS problem, associated with every point v i of the metric space is a cost ratio r i . We charge the online algorithm a cost of r i c i for dealing with the task (c 1 , . . . , c i , . . . , c n ) in state v i . which...

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The server problem and on-line games

Cited by 73 publications

References 19 publications

Reasoning about online algorithms with weighted automata

Reasoning about online algorithms with weighted automata

Randomized Competitive Analysis for Two Server Problems

Better Algorithms for Unfair Metrical Task Systems and Applications

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