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.

This paper proposes a protocol suite for dynamic replication and migration of Internet objects. It consists of an algorithm for deciding on the number and location of object replicas and an algorithm for distributing requests among currently available replicas. Our approach attempts to place replicas in the vicinity of a majority of requests while ensuring at the same time that no servers are overloaded. The request distribution algorithm uses the same simple mechanism to take into account both server proximity and load, without actually knowing the latter. The replica placement algorithm executes autonomously on each node, without the knowledge of other object replicas in the system. The proposed algorithms rely on the information available in databases maintained by Internet routers. A simulation study using synthetic workloads and the network backbone of UUNET, one of the largest Internet service providers, shows that the proposed protocol is effective in eliminating hot spots and ...

As commercial interest in the Internet grows, more and more companies are o#ering the service of hosting and providing access to information that belongs to third-party information providers. In the future, successful hosting services may host millions of objects on thousands of servers deployed around the globe. To provide reasonable access performance to popular resources, these resources will have to be mirrored on multiple servers. In this paper, we identify some challenges due to the scale that a platform for such global services would face, and propose an architecture capable of handling this scale. The proposed architecture has no bottleneck points. A trace-driven simulation using an access trace from AT&T's hosting service shows very promising results for our approach. Keywords: Hosting service, scalable architecture, dynamic replication, migration. 1 Introduction As commercial interest in the Internet grows, more and more companies are o#ering hosting services i.e. ...

Abstract We develop approximation algorithms for the problem of placing replicated data in arbitrary net-works, where the nodes may both issue requests for data objects and have capacity for storing data objects, so as to minimize the average data-access cost. We introduce the data placement problem tomodel this problem. We have a set of caches F, a set of clients D, and a set of data objects O. Each cache i can store at most ui data objects. Each client j 2 D has demand dj for a specific data object o(j) 2 O and has to be assigned to a cache that stores that object. Storing an object o in cache i incurs astorage cost of f oi, and assigning client j to cache i incurs an access cost of djcij. The goal is to find aplacement of the data objects to caches respecting the capacity constraints, and an assignment of clients

Consider a hierarchical network in which each node periodically issues a request for an object drawn from a fixed set of unit-size objects. Suppose further that the following conditions are satisfied: the frequency with which each node accesses each object is known; each node has a cache of known capacity; any cache can be accessed by any node; any request is satisfied by the closest node with a copy of the desired object, at a cost proportional to the distance between the accessing node and the closest copy. In such an environment, it is desirable to fill the available cache space with copies of objects in such a way that the average access cost is minimized. We provide both exact and approximate polynomial-time algorithms for this hierarchical placement problem. Our exact algorithm is based on a reduction to min-cost flow, and does not appear to be practical for large problem sizes. Thus we are motivated to search for a faster approximation algorithm. Our main result is a simple constant-factor approximation algorithm for the hierarchical placement problem that admits an efficient distributed implementation.

This paper introduces a framework for evaluating replica placement algorithms (RPA) for content delivery networks (CDN) as well as RPAs from other fields that might be applicable to current or future CDNs. First, the framework classifies and qualitatively compares RPAs using a generic set of primitives that capture problem definitions and heuristics. Second, it provides estimates for the decision times of RPAs using an analytic model. To achieve accuracy, the model takes into account disk accesses and message sizes, in addition to computational complexity and message numbers that have been considered traditionally. Third, it uses the "goodness" of produced placements to compare RPAs even when they have different problem definitions. Based on these evaluations, we identify open issues and potential areas for future research.

In this paper we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit, we present a (randomized) online algorithm with a competitive ratio of ln(p max min )+1, when bids are in a range [p min ,p max ], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. Interestingly, we show that if the online algorithm is allowed to subsidize matches --- match money-losing pairs if it has already collected enough money from previous pairs to pay for them --- then it can be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, the ability to subsidize is at least as valuable as knowing the future. We also consider the objectives of maximizing buy or sell volume, and present algorithms that achieve a competitive ratio of 2(ln(p max /p min ) + 1), or ln(p max /p min ) + 1 if the online algorithm is allowed subsidization. We show the latter is the best possible competitive ratio for this setting. For social welfare maximization we also obtain an optimal competitive ratio, which is below ln(p max /p min ). We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.

In this paper, we examine the bandwidth-constrained placement problem, focusing on trade-os appropriate for wide area network (WAN) environments. The goal is to place copies of objects at a collection of distributed caches to minimize expected access times from distributed clients to those objects subject to a maximum bandwidth constraint at each cache. We develop a simple algorithm to generate a bandwidth-constrained placement by hierarchically rening an initial per-cache greedy placement. We prove that this hierarchical algorithm generates a placement whose expected access time is within a constant factor of the optimal placement's expected access time. We then proceed to extend this algorithm to compute close to optimal placement strategies for dynamic environments.

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...