We prove that the work function algorithm for the k-server problem has competitive ratio at most 2k \Gamma 1. Manasse, McGeoch, and Sleator [24] conjectured that the competitive ratio for the k-server 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 k-server 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...

The paging problem is defined as follows: we are given a two-level memory system, in which one level is a fast memory, called cache, capable of holding k items, and the second level is an unbounded but slow memory. At each given time step, a request to an item is issued. Given a request to an item p,amiss occurs if p is not present in the fast memory. In response to a miss, we need to choose an item q in the cache and replace it by p. The choice of q needs to be made on-line, without the knowledge of future requests. The objective is to design a replacement strategy with a small number of misses. In this paper we use competitive analysis to study the performance of randomized on-line paging algorithms. Our goal is to show how the concept of work functions, used previously mostly for the analysis of deterministic algorithms, can also be applied, in a systematic fashion, to the randomized case. We present two results: we first show that the competitive ratio of the marking algorithm is ex...

. The page migration problem is to manage a globally addressed shared memory in a multiprocessor system. Each physical page of memory is located at a given processor, and memory references to that page by other processors incur a cost proportional to the network distance. At times the page may migrate between processors at cost proportional to the distance times D, a page size factor. The problem is to schedule movements on-line so that the total cost of memory references is within a constant factor c of the best off-line schedule. An algorithm that does so is called c-competitive. Black and Sleator gave 3-competitive deterministic on-line algorithms for uniform networks (complete graphs with unit edge lengths) and for trees with arbitrary edge lengths. No good deterministic algorithm is known for general networks with arbitrary edge lengths. We present randomized algorithms for the migration problem that are both simple and better than 3-competitiveagainst an oblivious adversary. We ...

This paper is concerned with the page migration (or file migration) problem [BS89] as part of a large class of on-line problems. The page migration problem deals with the management of pages residing in a network of processors. In the classical problem there is only one copy of each page which is accessed by different processors over time. The page is allowed to be migrated between processors. However a migration incurs higher communication cost than an access (proportionally to the page size). The problem is that of deciding when and where to migrate the page in order to lower access costs. A more general setting is the k-page migration where we wish to maintain k copies of the page. The page migration problems are concerned with a dilemma common to many on-line problems: determining when is it beneficial to make configuration changes. We deal with the relaxed task systems model which captures a large class of problems of this type, that can be described as the generalizati...

We present improved competitive on-line algorithms for the page replication problem and concentrate on important network topologies for which algorithms with a constant competitive ratio can be given. We develop an optimal randomized on-line replication algorithm for trees and uniform networks; its competitive ratio is approximately 1.58. This performance holds against oblivious adversaries. We also give a randomized memoryless replication algorithm for trees and uniform networks that is 2-competitive against adaptive on-line adversaries. Furthermore we consider on-line replication algorithms for rings and present general techniques that transform c-competitive algorithms for trees into 2c-competitive algorithms for rings. As a result we obtain a randomized on-line algorithm for rings that is 3.16-competitive. We also derive two 4-competitive on-line algorithms for rings which are either deterministic or randomized and memoryless. Again, the randomized results hold against oblivious ad...

. Competitive on-line algorithms for data management in a network of processors are studied in this paper. A data object such as a file or a page of virtual memory is to be read and updated by various processors in the network. The goal is to minimize the communication costs incurred in serving a sequence of such requests. Distributed data management on important classes of networks---trees and bus based networks, are studied. Optimal algorithms with constant competitive ratios and matching lower bounds are obtained. Our algorithms use different interesting techniques such as work functions [9] and "factoring." Key words. on-line algorithms, competitive analysis, memory management, data management. AMS subject classifications. 68Q20, 68Q25. 1. Introduction. The management of data in a distributed network is an important and much studied problem in management science, engineering, computer systems and theory [3, 11]. Dowdy and Foster [11] give a comprehensive survey of research in th...

The use of randomization in online multiprocessor scheduling is studied. The problem of scheduling independent jobs on m machines online originates with Graham [17]. While the deterministic case of this problem has been studied extensively, little work has been done on the randomized case. For m = 2 a randomized 4=3-competitive algorithm was found by Bartal, Fiat, Karloff and Vohra. A randomized algorithm for m 3 is presented. It achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295 and 1.81681, for m = 3; 4; 5; 6; 7 respectively. These competitive ratios are less than the best deterministic lower bound for m = 3; 4; 5 and less than the best known deterministic competitive ratio for m = 3; 4; 5; 6; 7. Two models of multiprocessor scheduling with rejection are studied. The first is the model of Bartal, Leonardi, Marchetti-Spaccamela, Sgall and Stougie. Two randomized algorithms for this model are presented. The first algorithm performs well for small m, achieving competitiv...

Contents iii Contents 1 Cover Page i 2 Project Summary ii 3 Table of Contents iii 4 Objectives and Aims 1 5 Narrative and Bibliography 2 5.1 Rationale of the Project . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.1.1 Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.1.2 Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.1.3 Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.2 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.2.1 NP-Hard Problems . . . . . . . . . . . . . . . . . . . . . . . . 4 5.2.2 Online Problems . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.2.3 Approximation Algorithms . . . . . . . . . . . . . . . . . . . . 6 5.2.4 Previous and Ongoing Work . . . . . . . . . . . . . . . . . . . 7 5.2.5 Planned Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2.6 Plans for Publication . . . . . . . . . . . . .

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