The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to low-distortion embeddings in low-dimensional 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 probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line al...

This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.

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...

Abstract The Sleator-Tarjan competitive analysis of paging [Comm. of the ACM; 28:202- 208, 1985] gives us the ability to make strong theoretical statements about the performance of paging algorithms without making probabilistic assumptions on the input. Nevertheless practitioners voice reservations about the model, citing its inability to discern between LRU and FIFO (algorithms whose performances differ markedly in practice), and the fact that the theoretical competitiveness of LRU is much larger than observed in practice. In addition, we would like to address the following important question: given some knowledge of a program's reference pattern, can we use it to improve paging performance on that program?

The competitive analysis of online algorithms has been criticized as being too crude and unrealistic. We propose refinements of competitive analysis in two directions: The first restricts the power of the adversary by allowingonly certain input distributions, while the other allows for comparisons between information regimes for online decision-making. We illustrate the first with an application to the paging problem; as a byproduct we characterize completely the work functions of this important special case of the k-server problem. We use the second refinement to explore the power of lookahead in server and task systems.

We deal with the competitive analysis of algorithms for managing data in a distributed environment. We deal with the file allocation problem ([DF], [ML]), where copies of a file may be be stored in the local storage of some subset of processors. Copies may be replicated and discarded over time so as to optimize communication costs, but multiple copies must be kept consistent and at least one copy must be stored somewhere in the network at all times. We deal with competitive algorithms for minimizing communication costs, over arbitrary sequences of reads and writes, and arbitrary network topologies. We define the constrained file allocation problem to be the solution of many individual file allocation problems simultaneously, subject to the constraints of local memory size. We give competitive algorithms for this problem on the uniform network topology. We then introduce distributed competitive algorithms for on-line data tracking (a generalization of mobile user tracking [AP1...

This paper deals with the file allocation problem [BFR92] concerning the dynamic optimization of communication costs to access data in a distributed environment. We develop a dynamic file re-allocation strategy that adapts on-line to a sequence of read and write requests whose location and relative frequencies are completely unpredictable. This is achieved by replicating the file in response to read requests and migrating the file in response to write requests while paying the associated communications costs, so as to be closer to processors that access it frequently. We develop first explicit deterministic on-line strategy assuming existence of global information about the state of the network; previous (deterministic) solutions were complicated and more expensive. Our solution has (optimal) logarithmic competitive ratio. The paper also contains the first explicit deterministic data migration [BS89] algorithm achieving the best known competitive ratio for this problem. Using somewhat ...

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...

Consider the on-line problem where a number of servers are ready to provide service to a set of customers. Each customer's job can be handled by any of a subset of the servers. Customers arrive one-by-one and the problem is to assign each customer to an appropriate server in a manner that will balance the load on the servers. This problem can be modeled in a natural way by a bipartite graph where the vertices of one side (customers) appear one at a time and the vertices of the other side (servers) are known in advance. We derive tight bounds on the competitive ratio in both deterministic and randomized cases. Let n denote the number of servers. In the deterministic case we provide an on-line algorithm that achieves a competitive ratio of k = dlog 2 ne (up to an additive 1) and prove that this is the best competitive ratio that can be achieved by any deterministic on-line algorithm. In a similar way we prove that the competitive ratio for the randomized case is k 0 = ln(n) (up to an a...