A polylog(<i>n</i>)-competitive algorithm for metrical task systems

Bartal, Yair; Blum, Avrim; Burch, Carl; Tomkins, Andrew

doi:10.1145/258533.258667

Cited by 78 publications

(115 citation statements)

References 14 publications

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“…1 In fact, an algorithm for the r -unfair ratio on the uniform metric space is at the heart of the best algorithm known for the MTS problem on arbitrary metric spaces (Bartal et al, 1997) (together with an approximation of arbitrary spaces by hierarchical ones (Bartal, 1996)). …”

Section: The Mts Problem and The Competitive Ratiomentioning

confidence: 99%

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Blum

Burch

2000

Machine Learning

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Abstract. The problem of combining expert advice, studied extensively in the Computational Learning Theory literature, and the Metrical Task System (MTS) problem, studied extensively in the area of On-line Algorithms, contain a number of interesting similarities. In this paper we explore the relationship between these problems and show how algorithms designed for each can be used to achieve good bounds and new approaches for solving the other. Specific contributions of this paper include:• An analysis of how two recent algorithms for the MTS problem can be applied to the problem of tracking the best expert in the "decision-theoretic" setting, providing good bounds and an approach of a much different flavor from the well-known multiplicative-update algorithms.• An analysis showing how the standard randomized Weighted Majority (or Hedge) algorithm can be used for the problem of "combining on-line algorithms on-line", giving much stronger guarantees than the results of Azar, Y., Broder, A., & Manasse, M. (1993). Proc ACM-SIAM Symposium on Discrete Algorithms (pp. 432-440) when the algorithms being combined occupy a state space of bounded diameter.• A generalization of the above, showing how (a simplified version of) Herbster and Warmuth's weight-sharing algorithm can be applied to give a "finely competitive" bound for the uniform-space Metrical Task System problem. We also give a new, simpler algorithm for tracking experts, which unfortunately does not carry over to the MTS problem.Finally, we present an experimental comparison of how these algorithms perform on a process migration problem, a problem that combines aspects of both the experts-tracking and MTS formalisms.

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Section: The Mts Problem and The Competitive Ratiomentioning

confidence: 99%

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Blum

Burch

2000

Machine Learning

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“…A randomized online algorithm A is called r-competitive if there exists some constant c such that for any task sequence σ, E[cost A (σ)] ≤ r · cost Opt (σ) + c, where cost A (σ) is the random variable of the cost for serving σ by A, and cost Opt (σ) is the optimal (offline) cost for serving σ. The current best online algorithm for the MTS problem in n-point metric spaces is O(log 2 n log log n) competitive [16,15] (an improvement of [5,3]). Both papers [5,16] actually solve the MTS problem for ultrametrics, and then reduce arbitrary metric spaces to ultrametrics using Theorem 1.…”

Section: Applicationsmentioning

confidence: 99%

“…The current best online algorithm for the MTS problem in n-point metric spaces is O(log 2 n log log n) competitive [16,15] (an improvement of [5,3]). Both papers [5,16] actually solve the MTS problem for ultrametrics, and then reduce arbitrary metric spaces to ultrametrics using Theorem 1. We next show that path embedding suffices: Proposition 1.…”

Section: Applicationsmentioning

confidence: 99%

Multiembedding of Metric Spaces

Bartal¹,

Mendel²

2004

SIAM J. Comput.

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Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size.We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log ∆ log log ∆) where ∆ is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Ω(log n) lower bound for all previous notions of embeddings.We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems.

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“…A powerful technique that has been successfully used recently in this context is to embed the given metric space in a simpler metric space such that the distances are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems [1,2,7,11,12,20]. Tree metrics are a very natural class of simple metric spaces since many algorithmic problems become tractable on them.…”

Section: Introductionmentioning

confidence: 99%

Distance Approximating Trees: Complexity and Algorithms

Dragan

Yan

2006

Lecture Notes in Computer Science

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Abstract. Let Δ ≥ 1 and δ ≥ 0 be real numbers. A treeThe distance (Δ, δ)-approximating tree problem asks for a given graph G to decide whether G has a distance (Δ, δ)-approximating tree. In this paper, we consider unweighted graphs and show that the distance (Δ, 0)-approximating tree problem is NP-complete for any Δ ≥ 5 and the distance (1, 1)-approximating tree problem is polynomial time solvable.

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A polylog(n)-competitive algorithm for metrical task systems

Cited by 78 publications

References 14 publications

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Multiembedding of Metric Spaces

Distance Approximating Trees: Complexity and Algorithms

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