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...

Time-indexed linear programming formulations have recently received a great deal of attention for their practical effectiveness in solving a number of single-machine scheduling problems. We show that these formulations are also an important tool in the design of approximation algorithms with good worst-case performance guarantees. We give simple new rounding techniques to convert an optimal fractional solution into a feasible schedule for which we can prove a constant-factor performance guarantee, thereby giving the first theoretical evidence of the strength of these relaxations. Specifically, we consider the problem of minimizing the total weighted job completion time on a single machine subject to precedence constraints, and give a polynomialtime (4 + ffl)-approximation algorithm, for any ffl ? 0; the best previously known guarantee for this problem was superlogarithmic. With somewhat larger constants, we also show how to extend this result to the case with release date constraints, ...

Abstract. In this paper, we provide a new class of randomized approximation algorithms for scheduling problems by directly interpreting solutions to so-called time-indexed LPs as probabilities. The most general model we consider is scheduling unrelated parallel machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. The crucial idea for these multiple machine problems is not to use standard list scheduling but rather to assign jobs randomly to machines (with probabilities taken from an optimal LP solution) and to perform list scheduling on each of them. For the general model, we give a (2+ e)-approximation algorithm. The best previously known approximation algorithm has a performance guarantee of 16/3 [HSW96]. Moreover, our algorithm also improves upon the best previously known approximation algorithms for the special case of identical parallel machine scheduling (performance guarantee (2.89 + e) in general [CPS+96] and 2.85 for the average completion time [CMNS97], respectively). A perhaps surprising implication for identical parallel machines is that jobs are randomly assigned to machines, in which each machine is equally likely. In addition, in this case the algorithm has running time O(nlogn) and performance guarantee 2. The same algorithm also is a 2-approximation for the corresponding preemptive scheduling problem on identical parallel machines. Finally, the results for identical parallel machine scheduling apply to both the off-line and the on-line settings with no difference in performance guarantees. In the on-line setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. 1

1 We prove a lower bound of Ω(log n / log log n) on the competitive ratio of any (deterministic or randomized) distributed algorithm for solving the mobile user problem introduced by Awerbuch and Peleg [5], on certain networks of n processors. Our lower bound holds for various networks, including the hypercube, any network with sufficiently large girth, and any highly expanding graph. A similar Ω(log n / log log n) lower bound is proved for the competitive ratio of the maximum job delay of any distributed algorithm for solving the distributed scheduling problem of Awerbuch, Kutten and Peleg [4] on any of these networks. The proofs combine combinatorial techniques with tools from linear algebra and harmonic analysis and apply, in particular, a generalization of the vertex isoperimetric problem on the hypercube, which may be of independent interest. 2 Footnotes for title page

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...

In this paper, we provide a new class of randomized approximation algorithms for parallel machine scheduling problems. The most general model we consider is scheduling unrelated machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. We introduce an LP relaxation in time-indexed variables for this problem. The crucial idea to derive approximation results is not to use standard list scheduling, but rather to assign jobs randomly to machines (by interpreting LP solutions as probabilities), and to perform list scheduling on each of them. Our main result is a (2 + e)--approximation algorithm for this general model which improves upon performance guarantee 16=3 due to Hall, Shmoys, and Wein. In the absence of nontrivial release dates, we get a (3=2 + e)--approximation. At the same time we prove corresponding bounds on the quality of the LP relaxation. A perhaps surprising implication for identical parallel machines is that jobs are ra...

In most communication networks, pairs of processors communicate by sending messages over a path connecting them. We present communication e cient protocols that quickly detect and locate any failure along the path. Whenever there is excessive delay inforwarding messages along the path, the protocols detect a failure (even when the delay is caused by maliciously-programmed processors). The protocols ensure optimal time for either message delivery or failure detection. We observe that the actual delivery time of a message over a link is usually much smaller than the apriori known upper bound D on that delivery time. The main contribution of the paper is the way tomodelandtakeadvantage of this observation. We introduce the notion of asynchronously early determinating protocols, as well as protocols that are asynchronously early terminating, i.e., time optimal in both worse case and typical cases. More precisely, we present a time complexity measure according to which one evaluates protocols both in terms of D and.Weobserve that asynchronously early termination is a form of competitiveness. The protocols presented here are asynchorously early terminating since they are time optimal both in terms of D and of. Previous communication e cient solutions were slow in the case where D. Weobserve that this is the most typical case. Preliminary reports of parts of the work reported here appeared in the proceedings of the ICCC 88

Resource contention is widely recognized as having a major impact on the performance of distributed algorithms. Nevertheless, the metrics that are commonly used to predict their performance take little or no account of contention. In this paper, we define two performance metrics for distributed algorithms that account for network contention as well as CPU contention. We then illustrate the use of these metrics by comparing four Atomic Broadcast algorithms, and show that our metrics allow for a deeper understanding of performance issues than conventional metrics. 1 Introduction Performance prediction and evaluation are a central part of every scientific and engineering activity, including the construction of distributed applications. Engineers of distributed systems rely heavily on various performance evaluation techniques and have developed the necessary techniques for this activity. In contrast, algorithm designers invest considerable effort in proving the correctness of their algor...

Scheduling a set of tasks on a set of machines so as to yield an efficient schedule is a basic problem in computer science and operations research. Most of the research on this problem incorporates the potentially unrealistic assumption that communication between the different machines is instantaneous. In this paper we remove this assumption and study the problem of network scheduling, where each job originates at some node of a network, and in order to be processed at another node must take the time to travel through the network to that node. Our main contribution is to give approximation algorithms and hardness proofs for fully general forms of the fundamental problems in network scheduling. We consider two basic scheduling objectives: minimizing the makespan, and minimizing the average completion time. For the makespan we prove small constant factor hardness-to-approximate and approximation results. For the average completion time, we give a log-squared approximation algorithm for...

We study the problem of balancing the load on processors of an arbitrary network. If jobs arrive or depart during the process of load balancing, we have the dynamic load balancing problem; otherwise, we have the static load balancing problem. While static load balancing on arbitrary and special networks has been well studied, very little is known about dynamic load balancing. The difficulty lies in modeling the arrivals and departures of jobs in a clean manner. In this paper, we initiate the study of dynamic load balancing by modeling job traffic using an adversary. Our main result is that a simple, local control distributed load balancing algorithm maintains the load of the network within a stable level against this powerful adversary. Our results hold for different models of traffic patterns and processor communication. 1 Introduction An important problem in a distributed system is to balance the total workload among the various processors of the underlying system. Such load balan...