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Distributed Approaches to Triangulation and Embedding
 In Proceedings 16th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2005
"... A number of recent papers in the networking community study the distance matrix defined by the nodetonode latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a lowdimensional Euclidean space. In such algorithms it ..."
Abstract

Cited by 30 (6 self)
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A number of recent papers in the networking community study the distance matrix defined by the nodetonode latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a lowdimensional Euclidean space. In such algorithms it is feasible to measure distances among only a linear or nearlinear number of node pairs; the rest of the distances are simply not available. Moreover, for applications it is desirable to spread the load evenly among the participating nodes. Indeed, several recent studies use this ’fully distributed ’ approach and achieve, empirically, a low distortion for all but a small fraction of node pairs. This is concurrent with the large body of theoretical work on metric embeddings, but there is a fundamental distinction: in the theoretical approaches to metric embeddings, full and centralized access to the distance matrix is assumed and heavily used. In this paper we present the first fully distributed embedding algorithm with provable distortion guarantees for doubling metrics (which have been proposed as a reasonable abstraction of Internet latencies), thus providing some insight into the empirical success of the recent Vivaldi algorithm [7]. The main ingredient of our embedding algorithm is an improved fully distributed algorithm for a more basic problem of triangulation, where the triangle inequality is used to infer the distances that have not been measured; this problem received a considerable attention in the networking community, and has also been studied theoretically in [19]. We use our techniques to extend ɛrelaxed embeddings and triangulations to infinite metrics and arbitrary measures, and to improve on the approximate distance labeling scheme of Talwar [36]. 1
Linear advice for randomized logarithmic space
 Electronic Colloquium on Computational Complexity
, 2005
"... Abstract. We show that RL t, L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of nonuniform advice. To prove our result we use an ultralow space walk on the GabberGalil expander graph due to Gutfreund and ..."
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Cited by 4 (0 self)
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Abstract. We show that RL t, L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of nonuniform advice. To prove our result we use an ultralow space walk on the GabberGalil expander graph due to Gutfreund and Viola. 1 Introduction The question of whether RL, randomized logarithmic space, can be simulated in L, deterministic logarithmic space, remains a central challenge in complexitytheory. The best known deterministic simulation of randomized logarithmic space is due to Saks and Zhou [16] who, building on seminal work due toNisan [13], proved that
Electronic Colloquium on Computational Complexity, Report No. 42 (2005) Linear Advice for Randomized Logarithmic Space
, 2005
"... We show that RL ⊆ L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of nonuniform advice. To prove our result we show how to take an ultralow space walk on the GabberGalil expander graph. Work done while a ..."
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We show that RL ⊆ L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of nonuniform advice. To prove our result we show how to take an ultralow space walk on the GabberGalil expander graph. Work done while at TTIChicago.