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13
Probabilistic Approximation of Metric Spaces and its Algorithmic Applications
 In 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized ..."
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Cited by 323 (28 self)
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The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilisticallyapproximates another metric space. We prove that any metric space can be probabilisticallyapproximated by hierarchically wellseparated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divideandconquer algorithmic approach. The technique presented is of particular interest in the context of online computation. A large number of online al...
On the kServer Conjecture
 Journal of the ACM
, 1995
"... We prove that the work function algorithm for the kserver problem has competitive ratio at most 2k \Gamma 1. Manasse, McGeoch, and Sleator [24] conjectured that the competitive ratio for the kserver problem is exactly k (it is trivially at least k); previously the best known upper bound was ex ..."
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Cited by 95 (6 self)
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We prove that the work function algorithm for the kserver problem has competitive ratio at most 2k \Gamma 1. Manasse, McGeoch, and Sleator [24] conjectured that the competitive ratio for the kserver problem is exactly k (it is trivially at least k); previously the best known upper bound was exponential in k. Our proof involves three crucial ingredients: A quasiconvexity property of work functions, a duality lemma that uses quasiconvexity to characterize the configurations that achieve maximum increase of the work function, and a potential function that exploits the duality lemma. 1 Introduction The kserver problem [24, 25] is defined on a metric space M, which is a (possibly infinite) set of points with a symmetric distance function d (nonnegative real function) that satisfies the triangle inequality: For all points x, y, and z d(x; x) = 0 d(x; y) = d(y; x) d(x; y) d(x; z) + d(z; y) 1 On the metric space M, k servers reside that can move from point to point. A possib...
Randomized Algorithms for Metrical Task Systems
 Theoretical Computer Science
, 1995
"... Borodin, Linial, and Saks introduce a general model for online systems in [BLS92] called task systems and show a deterministic algorithm which achieves a competitive ratio of 2n \Gamma 1 for any metrical task system with n states. We present a randomized algorithm which achieves a competitive ratio ..."
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Cited by 21 (2 self)
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Borodin, Linial, and Saks introduce a general model for online systems in [BLS92] called task systems and show a deterministic algorithm which achieves a competitive ratio of 2n \Gamma 1 for any metrical task system with n states. We present a randomized algorithm which achieves a competitive ratio of e=(e\Gamma1)n\Gamma1=(e\Gamma1) ß 1:5820n\Gamma0:5820 for this same problem. For the uniform metric space, Borodin, Linial, and Saks present an algorithm which achieves a competitive ratio of 2Hn , and they show a lower bound of Hn for any randomized algorithm. We improve their upper bound for the uniform metric space by showing a randomized algorithm which is \Gamma Hn +O( p log n) \Delta competitive. 1 Introduction In computer systems, it is often necessary to solve problems with incomplete information. The input evolves with time, and incremental computational decisions must be made based on only part of the input. A typical situation is where a sequence of tasks must be perfo...
Unfair Problems and Randomized Algorithms for Metrical Task Systems
, 1998
"... Borodin, Linial, and Saks introduce a general model for online systems in [Borodin et al. 1992] called metrical task systems. In this paper, the unfair two state problem, a natural generalization of the two state metrical task system problem, is studied. A randomized algorithm for this problem is pr ..."
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Cited by 13 (3 self)
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Borodin, Linial, and Saks introduce a general model for online systems in [Borodin et al. 1992] called metrical task systems. In this paper, the unfair two state problem, a natural generalization of the two state metrical task system problem, is studied. A randomized algorithm for this problem is presented, and it is shown that this algorithm is optimal. Using the analysis of unfair two state problem, a proof of a decomposition theorem similar to that of Blum, Karloff, Rabani and Saks [Blum et al. 1992] is presented. This theorem allows one to design divide and conquer algorithms for specific metrical task systems. Our theorem gives the same bounds asymptotically, but has less restrictive boundary conditions. 1 Introduction In computer systems, it is often necessary to solve problems with incomplete information. The input evolves with time, and incremental computational decisions must be made based on only part of the input. A typical situation is where a sequence of tasks must be pe...
A General Decomposition Theorem for the kServer Problem
 Information and Computation
, 2001
"... The first general decomposition theorem for the kserver problem is presented. Whereas all previous theorems are for the case of a finite metric with k + 1 points, the theorem given here allows an arbitrary number of points in the underlying metric space. This theorem implies O(polylog(k))compet ..."
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Cited by 9 (0 self)
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The first general decomposition theorem for the kserver problem is presented. Whereas all previous theorems are for the case of a finite metric with k + 1 points, the theorem given here allows an arbitrary number of points in the underlying metric space. This theorem implies O(polylog(k))competitive algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated subspaces. The only other cases for which polylogarithmic competitive algorithms are known are the uniform metric space, and the weighted cache metric space with two weights. 1 Introduction The kserver problem is one of the most intriguing problems in the area of online algorithms [9, 17]. Furthermore, it has as special cases several important and practical problems. The most prominent of these is weighted caching, which has applications in the management of web browser caches. We investigate the randomized variant of this problem, for which very few results are known. Our main result...
Online algorithms and the kserver conjecture
, 1994
"... The Work Function Algorithm, a natural algorithm for the kserver problem, is shown to have competitive ratio at most 2k − 1 for all metric spaces. It is also shown that the kserver conjecture, which states that there is an online algorithm for the kserver problem with competitive ratio k, holds ..."
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Cited by 5 (4 self)
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The Work Function Algorithm, a natural algorithm for the kserver problem, is shown to have competitive ratio at most 2k − 1 for all metric spaces. It is also shown that the kserver conjecture, which states that there is an online algorithm for the kserver problem with competitive ratio k, holds for all metric spaces with k + 2 points. Furthermore, two refinements of competitive analysis are proposed and studied: diffuse adversaries and comparative analysis. They address successfully some of the drawbacks of competitive analysis. Chapter 1
A PolylogarithmicCompetitive Algorithm for the kServer Problem
"... We give the first polylogarithmiccompetitive randomized online algorithm for the kserver problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log 2 k) for any metric space on n points. Our algorithm improves upon the deterministic (2k − ..."
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Cited by 3 (0 self)
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We give the first polylogarithmiccompetitive randomized online algorithm for the kserver problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log 2 k) for any metric space on n points. Our algorithm improves upon the deterministic (2k − 1)competitive algorithm of Koutsoupias and Papadimitriou [23] whenever n is subexponential in k.
What To Do With Your Free Time: Algorithms for Infrequent Requests and Randomized Weighted Caching
, 1996
"... We consider an extension of the standard online model to settings in which an online algorithm has free time between successive requests in an input sequence. During this free time, the algorithm may perform operations without charge before receiving the next request. For instance, in planning the ..."
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Cited by 2 (0 self)
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We consider an extension of the standard online model to settings in which an online algorithm has free time between successive requests in an input sequence. During this free time, the algorithm may perform operations without charge before receiving the next request. For instance, in planning the motion of fire trucks, there may be time in between fires that one could use to reposition the trucks in anticipation of the next fire. We prove both upper and lower bounds on the power of deterministic and randomized algorithms in this model. As our main lemma, we show an O(log 2 k)competitive algorithm and an\Omega\Gamma/44 k) lower bound on the competitive ratio for any weighted caching problem on (k + 1)point spaces in the standard online model, thus making progress on an open problem of [MMS88b, You91]. These results also apply to any metrical task system on spaces corresponding to weighted star graphs. We also consider extensions to the standard online model in which both free t...
Uniform Metrical Task Systems with a Limited Number of States Abstract
"... on a uniform space of k points, for any k ≥ 2, where Hk = P k i=1 for any metrical task system 1 i, the kth harmonic number. This algorithm has better competitiveness than the IraniSeiden algorithm if k is smaller than 10 8. The algorithm is better by a factor of 2 if k < 47. ..."
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on a uniform space of k points, for any k ≥ 2, where Hk = P k i=1 for any metrical task system 1 i, the kth harmonic number. This algorithm has better competitiveness than the IraniSeiden algorithm if k is smaller than 10 8. The algorithm is better by a factor of 2 if k < 47.
On Multithreaded Metrical Task Systems
, 1999
"... Traditionally, online problems have been studied under the assumption that there is a unique sequence of requests that must be served. This approach is common to most general models of online computation, such as Metrical Task Systems. However, there exist online problems in which the requests ar ..."
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Traditionally, online problems have been studied under the assumption that there is a unique sequence of requests that must be served. This approach is common to most general models of online computation, such as Metrical Task Systems. However, there exist online problems in which the requests are organized in more than one independent thread. In this more general framework, at every moment the first unserved request of each thread is available. Therefore, apart from deciding how to serve a request, at each stage it is necessary to decide which request to serve among several possibilities. In this paper we introduce Multithreaded Metrical Task Systems, that is, the generalization of Metrical Task Systems to the case in which there are many threads of tasks. We study the problem from a competitive analysis point of view, proving lower and upper bounds on the competitiveness of online algorithms. We consider finite and infinite sequences of tasks, as well as deterministic and ran...