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 focuses on three natural strategies in which the mobile users make the decisions when and where to update: the time-based strategy, the number of movements-based strategy, and the distance-based strategy. We consider both memoryless movement patterns and movements with Markovian memory along a topology of cells arranged as a ring. We analyze the performance of each one of the three strategies under such movements, and show the performance differences between the strategies.

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 present a two-level hierarchical routing architecture for ad hoc networks. Within each lower level cluster, we describe a self-organizing, dynamic spine structure to (a) propagate topology changes, (b) compute updated routes in the background, and (c) provide backup routes in case of transient failures of the primary routes. We analyze and bound the worst case of movements between upper level clusters to show that this hierarchical architecture scales well with network size. 1 Introduction Ad hoc networks are multihop networks in which mobile hosts share a scarce wireless channel. In ad hoc networks, the network topology changes frequently. Hence, routing algorithms must expend overhead either to maintain current routing and topology tables or to discover up-to-date routes. Currently, most routing algorithms for ad hoc networks are flat, that is, designed with only one level of hierarchy. These flat routing algorithms can suffer from excessive overhead as network sizes increase. I...

This paper investigates a zero-sum game played on a weighted connected graph G between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree T and the edge player chooses an edge e. The payoff to the edge player is cost(T; e), defined as follows: If e lies in the tree T then cost(T; e) = 0; if e does not lie in the tree then cost(T; e) = cycle(T; e)=w(e), where w(e) is the weight of edge e and cycle(T; e) is the weight of the unique cycle formed when edge e is added to the tree T. Our main result is that the value of the game on any n-vertex graph is bounded above by exp(O( p log n log log n)). The game arises in connection with the k-server problem on a road network; i.e., a metric space that can be represented as a multigraph G in which each edge e represents a road of length w(e). We show that, if the value of the game on G is V al(G; w), then there is a randomized strategy that achieves a competitive ratio of k(1 + V al(G; w)) against any oblivious adversary. Thus, on any n-vertex road network, there is a randomized algorithm for the k-server problem that is k exp(O( p log n log log n))-competitive against oblivious adversaries. At the heart of our analysis of the game is an algorithm that, for any n-vertex weighted, connected multigraph, constructs a spanning tree T such

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

This article focuses on routing messages along shortest paths in tree networks, using compact distributed data structures. We mainly prove that n-node trees support routing schemes with message headers, node addresses, and local memory space of size O(log n) bits, and such that every local routing decision is taken in constant time. This improves the best known routing scheme by a factor of O(log n) in term of both memory requirements and routing time. Our routing scheme requires headers and addresses of size slightly larger than log n, motivated by an inherent trade-off between address-size and memory space, i.e., any routing scheme with addresses on log n bits requires n) bits of local memory-space. This shows that a little variation of the address size, e.g., by an additive O(log n) bits factor, has a significant impact on the local memory space.

. We introduce the concept of a class of graphs being locally tree-decomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We show that for each locally tree-decomposable class C of graphs and for each property ' of graphs that is denable in rst-order logic, there is a linear time algorithm deciding whether a given graph G 2 C has property '. 1 Introduction It is an important task in the theory of algorithms to nd feasible instances of otherwise intractable algorithmic problems. A notion that has turned out to be extremely useful in this context is that of tree-width of a graph. 3-Colorability, Hamiltonicity, and many other NP-complete properties of graphs can be decided in linear time when restricted to graphs whose tree-width is bounded by a xed constant (see [Bod97] for a survey). Courcelle [Cou90] proved a meta-theorem, which easily implies numer...