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

Against an adaptive adversary, we show that the power of randomization in online algorithms is severely limited! We prove the existence of an efficient "simulation" of randomized online algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases. 1 Introduction and Overview of Results Beginning with the work of Sleator and Tarjan [17], there has recently been a development of what might be called a Theory of Online Algorithms. The particular algorithmic problems analyzed in the Sleator and Tarjan paper are "list searching" and "paging", both well studied problems. But the novelty of their paper lies in a new measure of performance, later to be called the "competitive ratio", for online algorithms. This new approach, called "competitive analysis" in Karlin, Manasse, Rudolph and Sleator [11], seems to have...

In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, the relative information of knowing the general direction of the goal is much higher than knowing the distance to the goal.

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...

A token located at some vertex v of a connected, undirected graph G on n vertices is said to be taking a "random walk" on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of v. We consider the following problem: suppose that two tokens are placed on G, and at each tick of the clock a certain demon decides which of them is to make the next move. The demon is trying to keep the tokens apart as long as possible. What is the expected time M before they meet? The problem arises in the study of self-stabilizing systems, a topic of recent interest in distributed computing. Since previous upper bounds for M were exponential in n, the issue was to obtain a polynomial bound. We use a novel potential function argument to show that in the worst case M = \Gamma 4 27 + o(1) \Delta n 3 . 1 Introduction Let G be a connected graph on n vertices, and let v be a fixed vertex of G. A random walk on G, beginning at v, is a stochastic process whose stat...

In this paper we give deterministic competitive k-server algorithms for all k and all metric spaces. This settles the k-server conjecture [MMS] up to the competitive ratio. The best previous result for general metric spaces was a 3-server randomized competitive algorithm [BKT] and a non-constructive proof that a deterministic 3-server competitive algorithm exists [BBKTW]. The competitive ratio we can prove is exponential in the number of servers. Thus, the question of the minimal competitive ratio for arbitrary metric spaces is still open. 1

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constant d. We prove lower bounds for deterministic list update algorithms and 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]. Ro...

We study scheduling problems in battery-operated computing devices, aiming at schedules with low total energy consumption. While most of the previous work has focused on finding feasible schedules in deadline-based settings, in this article we are interested in schedules that guarantee good response times. More specifically, our goal is to schedule a sequence of jobs on a variable-speed processor so as to minimize the total cost consisting of the energy consumption and the total flow time of all jobs. We first show that when the amount of work, for any job, may take an arbitrary value, then no online algorithm can achieve a constant competitive ratio. Therefore, most of the article is concerned with unit-size jobs. We devise a deterministic constant competitive online algorithm and show that

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...