A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In ACM STOC, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for the synthesis of such walks, and employ it to design competitive on line algorithms. IBM T.J. Watson Research Center, Yorktown Heights, NY 10598. AT T Bell Laboratories, Murray Hill, NJ 07974. 1 Overview Much recent work has dealt with the competitive analysis of on line algorithms [5, 16, 18]. In this paper we study the design of randomized on line algorithms. We show here that the synthesis of random walks on graphs with positive real costs on theirs edges is related to the design of these randomized on line algorithms. We develop methods for the synthesis of such random walks, and ....

....previous moves, and the next cat move depends only on its current position. Some special cases of the cat and mouse game have been studied by Baeza Yates et al. 1] We show that this cat and mouse game is at the core of many other on line algorithms that have evoked tremendous interest of late [3, 4, 5, 8, 9, 11, 18, 20, 21, 22]. We consider two settings. The first is the k server problem, defined in [18] An on line algorithm manages k mobile servers located at the vertices of a graph G whose edges have positive real lengths; it has to satisfy on line a sequence of requests for service at vertex v i , i = 1; 2; ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745--763, 1992.

....finds the mouse. This strategy is very simple, and memoryless: the cat need not remember its previous moves, and the next cat move depends only on its current position. We show that this cat and mouse game is at the core of many other online algorithms that have evoked tremendous interest of late [2, 3, 4, 7, 8, 10, 14, 16, 17, 18]. We consider two settings. The first is the k server problem, defined in [14] An on line algorithm manages k mobile servers located at the nodes of a graph G whose edges have positive real lengths; it has to satisfy a sequence of requests for service at node v i , i = 1; 2; by moving a ....

.... 1 nodes. This includes all previously known cases where the conjecture was proven true, as well as many new cases. We do so with a single unified theory that of resistive inverses. The algorithm is very simple, and memoryless. The other setting is the metrical task system (MTS) defined in [4]. A MTS consists of a weighted graph (the nodes of the graph are states, and edge weights are the costs of moving between states) The algorithm occupies one state at any time. A task is represented by a vector (c 1 ; c n ) where c i is the cost of processing the task in state i. The ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745--763, 1992.

....we may also need to incur some kind of movement cost (we will discuss this more fully in Section 3. 3) This notion of an on line algorithm having state, with a cost for moving between states, is captured by the Metrical Task Slstem (MTS) problem studied in the On line Algorithms literature [5, 12]. In this problem, we imagine the on line algorithm as controlling a system that can be in one of n states or configurations, with some distance metric d among the states specifying the cost of moving from one state to another. This system incrementally receives a sequence of tasks, where each ....

....of segments in P. We will seek to achieve this for a and a2 as small as possible. The easier goal of competing against the best single expert is a restriction of the partitioning bound to the case kp 1. 2.2. The MTS problem and the competitive ratio In the Metrical Task System (MTS) problem [5, 12], an on linc algorithm controls a system with n states located at points in a space with distance metric d. The algorithm receives, one at a time, a sequence of tasks, each of which has a cost vector specifying the cost of performing the task in each state. Say the system currently occupies state ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. J of He ACM, 39(4):745 763, 1992.

....B ela Bollob as The University of Memphis Memphis, TN 38152 bollobas msci.memphis.edu Manor Mendel Tel Aviv University Tel Aviv, Israel mendelma tau.ac. il Abstract This paper gives a nearly logarithmic lower bound on the randomized competitive ratio for the Metrical Task Systems model [BLS92] This implies a similar lower bound for the extensively studied K server problem. Our proof is based on proving a Ramsey type theorem for metric spaces. In particular we prove that in every metric space there exists a large subspace which is approximately a hierarchically well separated ....

....1 Introduction This paper deals with the analysis of the performance of randomized online algorithms in the context of two fundamental online problems, the problem of metrical task systems and the K server problem. A metrical task system (MTS) introduced by Borodin, Linial, and Saks [BLS92] is a system that may be in one of a set of n internal states. The aim of the system is to perform a certain sequence of tasks. The performance of each task has a certain cost that depends on the task and the state of the system. The system, also called the server, may switch states; the cost of ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks, An optimal online algorithm for metrical task systems, J. Assoc. Comput. Mach. 39 (1992), no. 4, 745-763.

....incur loss by selecting a bad expert, but it also incurs loss for moving between experts. This notion of an on line algorithm having state, with a cost for moving between states, is captured by a problem studied in the On line Algorithms literature called the Metrical Task System (MTS) problem [BLS92, IS95] In this problem, we imagine the on line algorithm is controlling a system that can be in one of n states or configurations. This system is given a sequence of tasks, where each task has a cost vector specifying the cost of performing the task in each state of the system. There is also a ....

....the other has an unbounded competitive ratio. Finally, in Section 5 we present an empirical comparison of these algorithms and others for the process migration problem. 2 Definitions and general relations 2. 1 The MTS problem and the competitive ratio In the Metrical Task System (MTS) problem [BLS92, IS95] an on line algorithm controls a system with n states located at points in a space with distance metric d. The algorithm receives, one at a time, a sequence of tasks, each a cost vector specifying the cost of performing the task in each state. Say the system currently occupies state i and, ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. J of the ACM, 39(4):745--763, 1992.

....case of paging, each request to a page of memory requires that page to be present in the cache; in the k server problem one server must be present in one speci ed location to serve the request. The problems that we consider are a special case of the more general notion of Metrical Task Systems [6] that allows to model a great variety of on line problems. However, the generality of that approach implies that the results that have been proved are rather negative results. In fact, competitive algorithms with small constant competitiveness coecients have been found for di erent versions of the ....

.... coecients have been found for di erent versions of the paging and the k server problem [7, 8, 10, 11, 12, 13, 15, 17, 18, 20] On the other side, a lower bound on the competitiveness of any on line algorithm for Metrical Task Systems has been proved that is linear in the number of di erent states [6]. For the problems considered in this paper, the number of states is exponential in the size of the cache. However, the competitiveness coecients that we obtain are polynomial in the size of the cache. Besides their theoretical interest, the problems that we consider are motivated by the memory ....

A. Borodin, N. Linial, and M. Saks, An optimal online algorithm for metrical task systems, Proc. 19th Annual ACM Symposium on Theory of Computing 373-382 (1987).

....of the competitive ratio decision criterion. Competitive analysis has been applied to a variety of classical problems in computer science, such as the kserver problem [ Koutsoupias and Papadimitriou, 1995 ] and paging [ Fiat et al. 1991 ] as well as to more general algorithmic problems [ Borodin et al. 1992 ] In all of these studies the environment that the agent acts in is non strategic, and therefore does not assume to follow any rational behavior. In this paper we extend the concept of competitive analysis to the context of multiagent systems. In a multi agent system the environment In fact, ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745--763, 1992.

....some requests and rejecting others with the goal of maximizing the total resource utilization. The requests or packets must be serviced online in these applications since the future arrival sequence is generally not known. We use competitive analysis to measure the quality of our algorithms [13, 11, 6]; we consider both deterministic as well as randomized algorithms. We consider the oblivious competitive ratio in which the input sequence is selected independently of the random choices of the algorithm [5] Table 1: Summary of Results. We require that k is a real number greater than 1 and ....

....Pr[J 2 oe] Delta jJ j, where the probability that oe occurs is with respect to the algorithm A, meaning the probability that A produces schedule oe given S. We let GA (S) J2S GA (J ) When the algorithm being studied is clear, we just use G(J) and G(S) We use standard competitive analysis [13, 11, 6] to evaluate our algorithms. J j J j Figure 1: In the left drawing J j blocks J i , and in the right one J j covers J i . If J j = J then by our definition of blocking, J j blocks J i . Namely, let oe be an optimal solution (constructed by a computationally unbounded off line ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Communications of the ACM, 39(4):745--763, 1992. 19

....we carry out our analysis assuming that position m coincides with position 0, that is, the encyclopedia is circular. This assumption eliminates move (3) and clarifies our arguments. We then show how to transfer our results back to the linear case. We analyze the competitiveness of our algorithms [3, 9, 11, 13]. That is, we compare the performance of an online algorithm against the performance of an optimal offline algorithm that sees all requests in advance. An algorithm is called c competitive if its cost on any sequence of n requests is at most O(1) greater than c times the offline algorithm s cost. ....

....no assumptions about the distribution or correlation of requests are made; however, it measures performance relative to what is achievable by an omniscient algorithm, rather than in absolute terms. A discretized version of our problem is an example of a task system as defined by Borodin et al. [3]. Borodin et al., however, study a more general model in which the costs of serving requests rather than just request locations may be chosen by an adversary, so their bounds have no nontrivial implications for our problem. Other related work includes a number of recent papers on server ....

A. Borodin, N. Linial, and M. Saks, An Optimal Online Algorithm for Metrical Task Systems, 19th ACM Symp. on Theory of Computing , 1987.

....out our analysis assuming that position m coincides with position 0, that is, the encyclopedia is circular. This assumption eliminates move (3) and clarifies our arguments. We then show how to transfer most of our results back to the linear case. We analyze the competitiveness of our algorithms [3, 11, 13, 16]. That is, we compare the performance of an online algorithm against the performance of an optimal offline algorithm that sees all requests in advance. An algorithm is called c competitive if its cost on any sequence of n requests is at most O(1) greater than c times the offline algorithm s cost. ....

....the distribution or correlation of requests are made; however, it measures performance relative to what is achievable by an omniscient algorithm, rather than in absolute terms. 2 1.1. Related Work A discretized version of our problem is an example of a task system as defined by Borodin et al. [3]. Borodin et al. however, study a more general model in which the costs of servicing requests rather than just request locations may be chosen by an adversary, so their bounds have no nontrivial implications for our problem. Other related work includes a number of recent papers on server ....

A. Borodin, N. Linial, and M. Saks, An Optimal Online Algorithm for Metrical Task Systems, 19th ACM Symp. on Theory of Computing , 1987, 373--382.

.... the use of comparative analysis by attacking the question of the power of lookahead in on line problems of the server type: If L is the class of all algorithms with lookahead , and L 0 is the class of on line algorithms, then we show that, in the very general context of metrical task systems [3] we have R(L 0 ; L ) 2 1; that is, the ratio is at most 2 1 for all metrical task systems, and it is exactly 4 E. KOUTSOUPIAS AND C. PAPADIMITRIOU 2 1 for some) while in the more restricted context of paging R(L 0 ; L ) minf 1; kg: 2. Di use Adversaries. The competitive ....

....algorithm for a metrical task system has lookahead if it can base its decision not only on the past, but also on the next requests. All on line algorithms with lookahead comprise the information regime L . Thus, L 0 is the class of all traditional on line algorithms. Metrical task systems [3] are de ned on some metric space M; a server resides on some point of the metric space and can move from point to point. Its goal is to process on line a sequence of tasks T 1 ; T 2 ; The server is free to move to any position before processimg a task, although it has to pay the distance. ....

A. Borodin, N. Linial, and M. E. Saks, An optimal online algorithm for metrical task systems., Proceedings 19th Annual ACM Symposium on Theory of Computing, (1987), pp. 373-82.

....the adversary is ignorant of the outcomes of the random choices made by the online algorithm this adversary is called oblivious. 10 Indeed, it is often the case that randomization (against an oblivious adversary) dramatically improves the competitive performance (see the classical results of [8] and [15] regarding metrical task systems and virtual memory management) As we shall later see randomization empowers the online player also in the search problem. 1.5. Competitive Analysis: A Discussion. The main attraction in using the competitive ratio for analyzing online algorithms is that ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745--763, 1992. For the conference version see [7]. Optimal Search and One-Way Trading Online Algorithms 139

....An O(log 2 n) competitive randomized algorithm for metrical task systems on n equally spaced points on the line. Key words. online algorithms, randomized algorithms AMS subject classi cations. 68W20, 68W25, 68W40 1. Introduction. Metrical task systems, introduced by Borodin, Linial, and Saks [10], 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 de ned on the state space, and then pays the service cost associated with the new ....

....is constrained to t the particulars of the problem. In this paper we consider the original de nition 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 the original paper [10], along with a matching lower bound for any metric space. The randomized competitive ratio for 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 (log n) 10, 14] ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks, An optimal online algorithm for metrical task systems, J. Assoc. Comput. Mach., 39 (1992), pp. 745-763.

....factor less than H k . The marking algorithm is strongly competitive (its competitive factor is H k ) if k = n Gamma 1, but it is not strongly competitive if k n Gamma 1. We describe another 2 algorithm, EATR, which is strongly competitive for the case k = 2. Borodin, Linial, and Saks [3] gave the first specific problem in which the competitive factor is reduced if the on line algorithm is allowed to use randomness. The problem they analyzed is the uniform task system. They presented a randomized algorithm for uniform task systems whose competitive factor is 2H n , where n is the ....

A. Borodin, M. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Submitted for publication.

....first requested node with the assumption that initially no servers reside on nodes, and, in response to any request, any server that has not yet served a request may be placed on the requested node at no cost. Following a number of authors (Sleator and Tarjan [ST85] Borodin, Linial, and Saks [BLS87] and Manasse, McGeoch, and Sleator [MMS90] we are interested in strategies that are competitive, that is, strategies that on any sequence incur a cost bounded by some constant times the minimum cost possible for that sequence. Formally, ffl r denotes an arbitrary sequence of requests. ffl X ....

A. Borodin, M. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proc. 19th Annual ACM Symp. on Theory of Computing, pages 373--382, May 1987. New York, NY. JACM, to appear.

....relative importance of minimizing the number of acknowledgments and minimizing the amount of extra latency introduced to the connection. The weighting parameter j 2 [0; 1] would likely be specified by a network manager. Since our algorithms run in an on line fashion, we use competitive analysis [75, 51, 17] to study them. That is, we provide guarantees of how much more our algorithms solutions cost (as measured by the objective functions) than an optimal solution. Specifically, we say an algorithm is c competitive if we can guarantee that its solution always costs at most c times the cost of an ....

....each oe i . For f max , this cost is linear in time with slope (1 Gamma j) For f sum , this cost is piecewise linear in time with slope (1 Gamma j)joe i j, where joe i j increases by 1 with each new arrival until an acknowledgment is sent. 6. 3 Related Work We use standard competitive analysis [75, 51, 17] to evaluate our algorithms. Specifically, let C opt be the cost (for acknowledgments and latency) of an optimal solution and let CA be the cost of the solution produced by on line algorithm A. Then we say that A is ff competitive if for all sequences of arrivals, CA ffC opt . We are not ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Communications of the ACM, 39(4):745--763, 1992.

....i . To service the request, we have to move only one server to the requested point of its space. We will call this problem the sum of two 1 server problems. The cnn problem is the special case where both metric spaces M 1 and M 2 are lines. More generally, let 1 ; 2 ; n be task systems [7] (not necessarily distinct) We can synthesize these task systems to get two new interesting on line problems: the sum and the product of 1 ; 2 ; The sum is the problem where we get requests (tasks) for each task system and we have to service only one of them. The product, on the other ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, (STOC '87) , pages 373--382, 1987.

....task systems, the obvious way to generalize the definition from paging is to say that an optimal off line algorithm could process all requests optimally, using only the first k of the n possible states. However, this is essentially the same as if only those k states existed, so the results in [4] show that the accommodating function is A(k) 2k Gamma 1. 5.1 Minimizing flow times on m identical machines As an example of a very different type of problem where the accommodating function can be applied, we have considered a scheduling problem: the problem of minimizing flow time in a ....

Allan Borodin, Nathan Linial, and Michael E. Saks. An Optimal OnLine Algorithm for Metrical Task Systems. Journal of the ACM, 39:745-- 763, 1992. 36

....which system resource(s) if any, should be allocated to service the current request. Two of the many examples of interesting problems which fall into this single request sequence model are the paging problem [28, 23, 15] and its generalizations, the k server and generic task system problems [23, 21, 5]. While the single request sequence model captures many important problems, there are many others which do not fall into this category, such as some operating system scheduling problems [18, 12, 13, 24] and some real time scheduling problems [20, 4, 11] In a typical problem, there is a single ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task system. Journal of the ACM, 39:745-763, 1992.

....To service the request, we have to move only one server to the requested point of its space. We will call this problem the sum of two 1 server problems. The cnn problem is the special case where both metric spaces M 1 and M 2 are lines. More generally, let 1 ; 2 ; n be task systems [7] (not necessarily distinct) We can synthesize these task systems to get two new interesting on line problems: the sum and the product of 1 ; 2 ; The sum is the problem where we get requests (tasks) for each task system and we have to service only one of them. The product, on the other ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, (STOC '87) , pages 373-382, 1987.

....in the past in various papers [5, 19, 27] none of which considers the ratio metric. The reader is referred to [20] for a comprehensive survey of the results in these papers. The ratio measure #(R, n) has close connections to the competitiveness measure used in the study of on line algorithms [6, 23, 31]; indeed, our problem resembles an on line setting in which the obstacles encountered by the robot form a sequence of requests, and we compare its total cost R(S) to the o# line cost d(S) It is therefore worth pointing out some key di#erences between the models: a) In the navigation problem, ....

....future. c) Competitive analysis deals with request sequences of arbitrary (possibly infinite) length, whereas here we have a fixed number of obstacles in the scene. Thus we cannot cast our navigation problem in a standard on line framework such as the server problem [23] or metrical task systems [6]. Nevertheless, the analogy with on line algorithms proves useful in the study of randomized navigation (section 8) 2. The wall problem. In this section, we consider scenes in which t is an infinite vertical wall at distance n to the east of s and the obstacles are rectangles whose sides are ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks, An optimal online algorithm for metrical task systems, in Proc. 19th Annual ACM Symposium on Theory of Computing, ACM, New York, 1987, pp. 373--382.

....Chrobak and Larmore [4] For the case where we have a # Address: Department of Computer Science, 298 Coates Hall, Louisiana State University, Baton Rouge, LA 70803. Email: sseiden acm.org 1 finite metric with k 1 points, the problem is closely related to the metrical task system (MTS) problem [10, 3, 16]. The results on that problem imply that there is a O(polylog(k) competitive algorithm for every finite space with k 1 points. A O(log k) competitive algorithm for the weighted cache problem with 2 weights has recently been exhibited by Irani [18] In summary, the only two metrics for which a ....

Borodin, A., Linial, N., and Saks, M. An optimal online algorithm for metrical task systems. Journal of the ACM 39, 4 (Oct 1992), 745--763. 10

....outperform traditional algorithms for routing virtual circuits in ATM networks. See also Plotkin [1995] As customary, we evaluate the performance of the on line algorithms in terms of competitive ratio, introduced in Sleator and Tarjan [1985] and further developed in Karlin et al. 1988] Borodin et al. 1992], and Manasse et al. 1988] In our case, it corresponds to the supremum, over all possible input sequences, of the ratio of the maximum relative load achieved by the on line algorithm to the maximum relative load achieved by the optimal off line algorithm. Using our framework, we derive on line ....

BORODIN, A., LINIAL, N., AND SAKS, M. 1992. An optimal online algorithm for metrical task systems. J. ACM 39, 4 (Oct.), 745--763.

....it is dicult to show that the game corresponding to an online problem satis es von Neumann s minimax inequalities, making Theorem 1 inapplicable. There is, however, a version of Yao s principle for online problems having in nitely many input sequences that does not require the minimax inequalities [5]. Theorem 2 (Yao s principle, version 2) Let P be an online problem. Let f 1 ; 2 ; g denote the set of input sequences for P and let falg 1 ; alg 2 ; g denote the set of deterministic algorithms for P. Suppose that for all suciently large integers n, there exists a probability ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of hte ACM, 39:745-763, 1992.

....possible randomized algorithm. In practice this approach seems to easier than the alternative: devising an adversarial strategy against the set of all randomized algorithms. This principle was rst applied within the context of competitive analysis of online algorithms by Borodin, Linial and Saks [8]. Since that time it has repeatedly shown its usefulness in this area. However, despite the fact that many proofs via the von Neumann Yao principle share a common structure, beyond this principle there are no general tools to aid researchers. In this paper, it is our goal to try to rectify this ....

Borodin, A., Linial, N., and Saks, M. An optimal online algorithm for metrical task systems. Journal of the ACM 39, 4 (Oct 1992), 745-763.

....nature, allows much better theoretical bounds. 1 Machine learning theory and competitive analysis are two areas that have studied such distribution free problems. This thesis studies two fundamental problems in these areas: prediction from expert advice and metrical task systems [LW94, BLS92] The goal of the thesis is to explore the relationship between these two problems, with the specific aim of exploiting past ideas to achieve improved results for these problems and their variants. By relating and studying prediction from expert advice and metrical task systems, this thesis ....

....P . 2. 2 Metrical task systems The metrical task system (MTS) problem, proposed by Borodin, Linial, and Saks, is fundamental to competitive analysis in the same sense that the expert prediction problem is fundamental to on line learning theory: It abstracts many important on line problems [BLS92] 2.2.1 Problem statement Say we are controlling a system of n states with a distance metric d between them. At all times our online algorithm occupies a state. Each time step, we receive a task vector, specifying a cost in each state (representing the cost to process the task in each state) ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. J of the ACM, 39(4):745--763, 1992.

....This, combined with the result of [Bar96] yields the general bound. Note that for the k server problem on metric spaces of k c points our result implies a competitive ratio of O(c 6 log 6 k) 1 Introduction The Metrical Task System (MTS) problem, introduced by Borodin, Linial, and Saks [BLS92] can be stated as follows. Consider a machine that can be in one of n states or configurations. This machine is given a sequence of tasks, where each task has an associated cost vector specifying the cost of performing the task in each state of the machine. There is also a distance metric among ....

....since the tasks can be viewed as generated by an adversary that produces the task sequence before any of A s random choices. Specifically, we say that randomized algorithm A has competitive ratio r if, for some a, for all oe, E[c A (oe) r Delta c OPT (oe) a Borodin, Linial, and Saks [BLS92] present a deterministic on line MTS algorithm that for any metric space achieves a competitive ratio of 2n Gamma 1 and prove that this is optimal for deterministic algorithms (in a strong sense: namely, there is no metric space for which a deterministic algorithm can guarantee a better ratio) ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. JACM, 39(4):745--763, 1992.

.... competitive randomized online algorithm [49] and Achlioptas, Chrobak and Noga [1] describe a simpler H k competitive randomized online algorithm using O(k 2 ) time per request. Lower bounds are usually derived using a game theoretic result, observed by Yao [59] and by Borodin, Linial and Saks [16]. For a probability distribution P on the request sequences, a (deterministic) online algorithm A is called c competitive for P, if E P [Cost A (oe) c Delta E P [Cost OPT (oe) b : for some constant b. Let c P A denote the infimum of all such c, then inf RA c OBL RA = sup P inf A c ....

Allan Borodin, Nathan Linial, and Michael Saks. An optimal online algorithms for metrical task systems. Volume 19 of Symposium on Theory of Computing, pages 373--382. ACM, 1987.

....in computer science. The dynamic nature of these problems demands algorithms for these problems to make decisions without full knowledge of the impact of their decisions on the future performance of these algorithms. Many data structuring problems and scheduling problems are on line problems. [5] proposed an abstract model for capturing on line problems, and subsequently [18] formulated a variant of on line problems the k server problem. k server problems can be used to model a variety of problems in computer science including, among others, paging and planning the motion of the ....

....the k server problem and some related notions in Section 2. In Section 2.2 we present a literature survey of the work that has been done on this problem. As will be seen there, the k server conjecture remains unresolved. Our work, presented in Sections 3, is, in a lot of ways, along the lines of [5] and [18] mentioned above. In Section 3 we focus on k server algorithms, that use a constant amount of scratch space, for the k server problem (we call these space bounded algorithms) in finite metric spaces. Most real life problems that can be modeled as k server problems get requests from a ....

[Article contains additional citation context not shown here]

Allan Borodin, Nathan Linial, and Michael Saks. An optimal online algorithm for metrical task systems. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 373--382, 1987.

....ratio. This follows from the lower bound on the competitive ratio of paging algorithms. So far, the truth of the k server conjecture has not been resolved, but a good candidate for a deterministic algorithm with ratio k is the work function algorithm, developed independently by several researchers [Borodin et al. 1987], Chrobak and Larmore, 1992] Manasse et al. 1988] The work function algorithm keeps track of the optimal off line costs and tries to make moves that keep its cost close to the optimal cost. Given a sequence of m tasks, the optimal off line cost can be computed with a simple dynamic ....

....space, and there are m requests, the flow instance consists of a network of size O(k(m n) Using a standard network flow algorithm [Tarjan, 1983] a solution to the off line problem can be found in time O(k 2 (m n) 2 log(k(m n) 5. 2 Metrical Task Systems Borodin, Linial and Saks [Borodin et al. 1987] proposed a more general model of on line computation called metrical task systems. A metrical task system consists of a set of n states S and a distance function ffi so that (S; ffi) is a metric space. At any time, the system must be in exactly one of the n states. A sequence oe of tasks is ....

[Article contains additional citation context not shown here]

Borodin, A., Linial, N., and Saks, M. (1987). An optimal online algorithm for metrical task systems. In Proc. 19th Annual ACM Symposium on Theory of Computing, pages 373--382.

....each oe i . For f max , this cost is linear in time with slope (1 Gamma j) For f sum , this cost is piecewise linear in time with slope (1 Gamma j)joe i j, where joe i j increases by 1 with each new arrival until an acknowledgment is sent. 6. 3 Related Work We use standard competitive analysis [75, 51, 17] to evaluate our algorithms. Specifically, let C opt be the cost (for acknowledgments and latency) of an optimal solution and let CA be the cost of the solution produced by on line algorithm A. Then we say that A is ff competitive if for all sequences of arrivals, CA ffC opt . We are not ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Communications of the ACM, 39(4):745--763, 1992.

....This, combined with the result of [Bar96] yields the general bound. Note that for the k server problem on metric spaces of k c points our result implies a competitive ratio of O(c 6 log 6 k) 1 Introduction The Metrical Task System (MTS) problem, introduced by Borodin, Linial, and Saks [BLS92] can be stated as follows. Consider a machine that can be in one of n states or configurations. This machine is given a sequence of tasks, where each task has an associated cost vector specifying the cost of performing the task in each state of the machine. There is also a distance metric among ....

....measure since the tasks can be viewed as generated by an adversary that produces the task sequence before any of A s random choices. Specifically, we say that randomized algorithm A has competitive ratio r if, for some a, for all oe, E[cA (oe) r Delta c OPT (oe) a Borodin, Linial, and Saks [BLS92] present a deterministic on line MTS algorithm with a competitive ratio of 2n Gamma 1 and prove that this is optimal for deterministic algorithms. They also show that with randomization one can achieve a competitive ratio of O(logn) for the special case of the uniform metric space. Several ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. JACM, 39(4):745--763, 1992.

.... on (k 1) point spaces (also known Cat and Mouse problems or Pursuit Evasion games [BKRS92] We also give an Omega Gamma 39 k) lower bound on the competitive ratio of any such algorithm, extending and simplifying results of [KRR91] Our results on weighted caching also apply to task systems [BLS87] Informally, a task system is specified as a set of N points with a distance function, as in server problems, but each request is an N vector r that may be served in any state j by paying the cost to move to point j and then paying the state specific cost r j . We give an O(log 2 ....

....or not free time is available. Prior results for general task systems include an Omega Gamma 12 N= log log N) lower bound of [BKRS92] and an en= e Gamma 1) competitive randomized algorithm due to Irani and Seiden [IS95] Results on specific spaces have focused largely on the uniform space: BLS87] in the journal version of their paper) give a 2HN upper bound and an HN lower bound. IS95] give an HN O( p log N) competitive algorithm, matching the lower bound to within lower order terms. 2 Definitions and Preliminaries An algorithm for the k server problem on a (k 1) point metric ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In 19th ACM Symposium on Theory of Computing, pages 373--382, 1987.

....for randomized algorithms against oblivious and adaptive on line adversaries. In particular, we show that for this problem adaptive on line and adaptive off line adversaries are equally powerful. 1 Introduction Recently much attention has been given to competitive analysis of on line algorithms [7, 20, 22, 25]. Roughly speaking, an on line algorithm is c competitive if, for any request sequence, its cost is no more than c times the cost of the optimum off line algorithm for that sequence. In their seminal work on competitive analysis [25] Sleator and Tarjan studied heuristics commonly used in system ....

....satisfied, and then any number of free exchanges are done. An on line list update algorithm must service each request without any knowledge of future requests. An off line algorithm is shown the entire sequence in advance; the optimum cost can always be achieved by an off line algorithm. Following [7, 20] we say a deterministic list update algorithm, A, is c competitive if there is a constant b such that for all size lists and all request sequences oe, A(oe) c Delta OPT(oe) b: For randomized list update algorithms, competitiveness is defined with respect to the model of an adversary. Two ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proc. 19th ACM Symposium on Theory of Computing, pages 373--382, 1987.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proc. 19th Annual ACM Symposium on Theory of Computing, pp. 373-382, 1987.

....when one also considers memory requirements, this greater generality is useful and we allow this generality in the definition. 3 We henceforth drop the superscript (V ; d) whenever it is clear from the context. We should note that an equivalent definition of competitive ratio as defined in [BLS87] and [MMS90] is given as the inffaej there exists a constant fi such that for all finite request sequences oe, CA (oe) Gamma ae Delta COPT (oe) fig. 4 if its competitive ratio is bounded by some constant. One notes that the optimal cost is well defined since oe is a finite sequence. ....

....fi such that for all finite request sequences oe, CA (oe) Gamma ae Delta COPT (oe) fig. 4 if its competitive ratio is bounded by some constant. One notes that the optimal cost is well defined since oe is a finite sequence. Moreover, dynamic programming affords an obvious algorithm (see [BLS87], MMS90] to realize the optimal bound. However, dynamic programming is an offline algorithm in that the entire sequence must be seen in order to produce the configuration (or server move) sequence. A much more efficient but still offline algorithm can be found in [CKPV90] An algorithm A is ....

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In 19th Annual ACM Symposium on Theory of Computing, pages 373--382, New York City, NY, May 1987. To appear in JACM.

....exchanges between elements that are not adjacent[9] 10] These alternative cost models can lead to qualitatively different results. 1. 2 Metrical task systems The (static) list update problem can also be considered within the metrical task system framework introduced by Borodin, Linial and Saks [8]. Metrical task systems (MTS) are an abstract model for online computation that captures a wide variety of online problems (paging, list update and the k server problem, to name a few) as special cases. A metrical task system is a system with n states, with a distance function d defined on the ....

....by the lower reference cost, this model is identical to the standard model. See [6] 3 One of the initial results about metrical task systems was that the work function algorithm (WFA) has competitive ratio 2n Gamma 1 for all MTS s, where n is the number of states in the metrical task system [8]. It was also shown that this is best possible, in the sense that there exist metrical task systems for which no online algorithm can achieve a competitive ratio lower than 2n Gamma 1. However, for many MTS s the upper bound of 2n Gamma 1 is significantly higher than the best achievable ....

[Article contains additional citation context not shown here]

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 52:46--52, 1985.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In ACM STOC, 1987.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task system. Journal of the ACM, 39:745--763, 1992.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task system. Journal of the ACM, 39:745-763, 1992.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task system. Journal of the ACM, 39:745-763, 1992.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745--763, 1992.

No context found.

Allan Borodin, Nathan Linial, and Michael Saks, An optimal online algorithm for metrical task system, Journal of the ACM 39 (1992), 745--763.

No context found.

A. Borodin, N. Linial, and M. Saks. An Optimal OnLine Algorithm for Metrical Task Systems. In Proc. of the 19th Ann. ACM Symp on Theory of Computing, pages 373{ 382, May 1987.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. J. Assoc. Comput. Mach., 39(4):745-763, 1992.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. Journal of the ACM, 39:745-763, 1992. 25

No context found.

A. Borodin, N. Linial, and M. Saks. "An Optimal Online Algorithm for Metrical Task Systems" Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, pp. 373--382.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proc. 19th Annual ACM Symposium on Theory of Computing, pages 373--382, 1987.

No context found.

A. Borodin, N. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pages 373--382, 1987. For the journal version see [8].

No context found.

A. Borodin, M. Linial, and M. Saks. An optimal online algorithm for metrical task systems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pages 373--382, New York, 1987.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC