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.

We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use\Omega\Gamma n) local space at some vertex. 1 Introduction Let G = (V; E) with jV j = n be a labeled undirected network. Assuming that a positive cost, or distance is assigned with each edge, the stretch of path p(u; v) from node u to node v is defined as jp(u;v)j jd(u;v)j , where jd(u; v)j is the length of the shortest u \Gamma v path. The approximate all-pairs shortest path problem involves a tradeoff of stretch against time-- short paths with stretch bounded by a constant are com...

This paper presents a family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information. The hierarchical schemes H k (for every integer k 1) guarantee a stretch factor of O(k 2 ) on the length of the routes and require storing at most O(kn 1 k log n log D) bits of routing information per vertex in an n-processor network with diameter D. The schemes are nameindependent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k. Key words: Communication networks, routing tables, communication-space trade-offs, graph covers. Dept. of Mathematics and Lab. for Computer Science, M.I.T., Cambridge, MA 02139; ARPANET: baruch@theory.lcs.mit.edu. Supported by Air Force Contract TNDGAFOSR-86-0078, ARO contract DAAL03-86-K-0171, NSF contract CCR8611442, DARPA contract N00014-89-J-1988, and a special grant from IBM. y Department of Applied Mathemati...

How to represent a graph in memory is a fundamental data structuring question. In the usual representations of an n-vertex graph, the names of the vertices (i.e. integers from 1 to n) betray nothing about the graph itself. Indeed, the names (or labels) on the n vertices are just log n bit place holders to allow data on the edges to encode the structure of the graph. In our scenario, there is no such waste. By assigning O(log n) bit labels to the vertices, we completely encode the structure of the graph, so that given the labels of two vertices we can test if they are adjacent in time linear in the size of the labels. Furthermore, given an arbitrary original labeling of the vertices, we can find structure coding labels (as above) that are no more than a small constant factor larger than the original labels. These notions are intimately related to vertex induced universal graphs of polynomial size. For example, we can label planar graphs with structure coding labels of size ! 4 log n, which implies the existence of a graph with n 4 vertices that contains all n-vertex planar graphs as vertex induced subgraphs.

This survey concerns the role of data structures for compactly storing and representing various types of information in a localized and distributed fashion. Traditional approaches to data representation are based on global data structures, which require access to the entire structure even if the sought information involves only a small and local set of entities. In contrast, localized data representation schemes are based on breaking the information into small local pieces, or labels, selected in a way that allows one to infer information regarding a small set of entities directly from their labels, without using any additional (global) information. The survey focuses on combinatorial and algorithmic techniques, and covers complexity results on various applications, including compact localized schemes for message routing in communication networks, and adjacency and distance labeling schemes.

This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, name-independent routing and labeled routing. The key results obtained for this model are the following. First, we provide the first name-independent solution. Specifically, we achieve constant stretch and polylogarithmic storage. Second, we obtain the first truly scale-free solutions, namely, the network’s aspect ratio is not a factor in the stretch. Scale-free schemes are given for three problem models: name-independent routing on graphs, labeled routing on metric spaces, and labeled routing on graphs. Third, we prove a lower bound requiring linear storage for stretch < 3 schemes. This has the important ramification of separating for the first time the name-independent problem model from the labeled model for these networks, since compact stretch-1+ε labeled schemes are known to be possible.

This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems.

Given a weighted undirected network with arbitrary node names, we present a compact routing scheme, using a O(&radic;n) space routing table at each node, and routing along paths of stretch 3, that is, at most thrice as long as the shortest paths. This is optimal in a very strong sense. It is known that no compact routing using o(n) space per node can route with stretch below 3. Also, it is known that any stretch below 5 requires &Omega;(&radic;n) space per node.

Several methods exist for routing messages in a network without using complete routing tables (compact routing). In k-interval routing schemes (k-IR.S), links carry up to k intervals each. A message is routed over certain link if its destination belongs to one of the intervals of the link. We give some results for the necessary value of k in order to achieve shortest path routing. Even though for very structured networks low values of suce, we show that for 'general graphs' interval routing cannot significantly reduce the space-requirements for shortest path routing. In particular we show that for suitably large n, there are suitable values of p such that for randomly chosen graphs G 6 ,p the following holds, with high probability: if G admits an optimal k-IIS, then k = The result is obtained by means of a novel matrix representation for the shortest paths in a network.