Abstract. In practice, almost all dynamic systems require decisions to be made on-line, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general on-line decnion algorithm is developed. It is shown that, for an important algorithms. class of special cases, this algorithm is optimal among all on-line Specifically, a task system (S. d) for processing sequences of tasks consists of a set S of states and a cost matrix d where d(i, j) is the cost of changing from state i to state j (we assume that d satisfies the triangle inequality and all diagonal entries are f)). The cost of processing a given task depends on the state of the system. A schedule for a sequence T1, T2,..., Tk of tasks is a ‘equence sl,s~,..., Sk of states where s ~ is the state in which T ’ is processed; the cost of a schedule is the sum of all task processing costs and state transition costs incurred. An on-line scheduling algorithm is one that chooses s, only knowing T1 Tz ~.. T’. Such an algorithm is w-competitive if, on any input task sequence, its cost is within an additive constant of w times the optimal offline schedule cost. The competitive ratio w(S, d) is the infimum w for which there is a w-competitive on-line scheduling algorithm for (S, d). It is shown that w(S, d) = 2 ISI – 1 for eoery task system in which d is symmetric, and w(S, d) = 0(1 S]2) for every task system. Finally, randomized on-line scheduling algorithms are introduced. It is shown that for the uniform task system (in which d(i, j) = 1 for all i, j), the expected competitive ratio w(S, d) =

The paging problem is that of deciding which pages to keep in a memory of k

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...

In the k-server problem, we must choose how k mobile servers will serve each of a sequence of requests, making our decisions in an online manner. We exhibit an optimal deterministic online strategy when the requests fall on the real line. For the weighted-cache problem, in which the cost of moving to x from any other point is w(x), the weight of x, we also provide an optimal deterministic algorithm. We prove the nonexistence of competitive algorithms for the asymmetric two-server problem, and of memoryless algorithms for the weighted-cache problem. We give a fast algorithm for offline computing of an optimal schedule, and show that finding an optimal offline schedule is at least as hard as the assignment problem. 1 Introduction The k-server problem can be stated as follows. We are given a metric space M , and k servers which move among the points of M , each occupying one point of M . Repeatedly, a request (a point x 2 M) appears. To serve x, each server moves some distance, possibly...

We study the design and analysis of randomized on-line algorithms.

Consider the following file caching problem: in response to a sequence of requests for files, where each file has a specified size and retrieval cost, maintain a cache of files of total size at most some specified k so as to minimize the total retrieval cost. Specifically, when a requested file is not in the cache, bring it into the cache, pay the retrieval cost, and choose files to remove from the cache so that the total size of files in the cache is at most k. This problem generalizes previous paging and caching problems by allowing objects of arbitrary size and cost, both important attributes when caching files for world-wide-web browsers, servers, and proxies. We give a simple deterministic on-line algorithm that generalizes many well-known paging and weighted-caching strategies, including least-recently-used, first-in-first-out,

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...

Weighted caching is a generalization of paging in which the cost to

We introduce and study a novel genre of optimization problems, which we call segmentation problems. Our motivation, in part, is the development of a framework for evaluating certain data mining and clustering operations in terms of their utility in decision-making. For any classical optimization problem, the corresponding segmentation problem seeks to partition a set of cost vectors into several segments, so that the overall cost is optimized. This framework contains a number of standard combinatorial clustering problems as special cases, and many segmentation problems turn out to be MAXSNP-complete even when the corresponding "un-segmented" version is easy to solve. We develop approximation algorithms for two natural and interesting problems in this class --- the HYPERCUBE SEGMENTATION PROBLEM and the CATALOG SEGMENTATION PROBLEM --- and present a general greedy scheme, which can be specialized to approximate a large class of segmentation problems. Finally, we indicate some connection...

We investigate the k-server problem when the metric space is a tree. For this case we present an on-line k-competitive algorithm for k servers. The competitiveness ratio k is optimal. The algorithm is memoryless, in the sense that it does not use any information from the past. 1 Introduction Let M be a metric space. That is, for any two points x; y 2 M we are given their distance kx; yk 0 such that kx; yk ? 0 for x 6= y, and the triangle inequality is also satisfied: kx; yk + ky; zk kx; zk for all x; y; z 2 M . We are also given k servers that can move among points of M . At each time slot, a request x 2 M appears, and we have to "serve" this request, that is, choose one of our servers and move it to x. Other servers are also allowed to move. Our measure of cost is the distance by which we move our servers. The problem is to design a strategy that minimizes the cost of servicing a sequence of requests given on-line. The server problem is an abstraction of several practical problem...