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18
On SpaceStretch TradeOffs: Lower Bounds
, 2006
"... One of the fundamental tradeoffs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the ratio between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. Using ..."
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Cited by 17 (6 self)
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One of the fundamental tradeoffs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the ratio between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. Using a distributed Kolmogorov Complexity argument, we give a lower bound for the nameindependent model that applies even to singlesource schemes and does not require a girth conjecture. For any integer k ≥ 1 we prove that any routing scheme for networks with arbitrary weights and arbitrary node names (even a singlesource routing scheme) with maximum stretch strictly less than 2k + 1 requires Ω((n log n) 1/k)bit routing tables. We extend our results to lower bound the averagestretch, showing that for any integer k ≥ 1 any nameindependent routing scheme with (n/(9k)) 1/kbit routing tables has averagestretch of at least k/4 + 7/8. This result is in sharp contrast to recent results on the averagestretch of labeled routing schemes.
Nearly tight low stretch spanning trees
 In Proceedings of the 49th Annual Symposium on Foundations of Computer Science
, 2008
"... We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT(u, v)/dG(u, v)] is bounded by Õ(log n). Our result is obtained via a new approach of building “highways ” between portals and a new strong diameter probabilis ..."
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Cited by 17 (0 self)
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We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT(u, v)/dG(u, v)] is bounded by Õ(log n). Our result is obtained via a new approach of building “highways ” between portals and a new strong diameter probabilistic decomposition theorem. 1
Embeddings of surfaces, curves, and moving points in euclidean space
 In Proc. 23rd Annu. ACM Sympos. Comput. Geom
, 2007
"... In this paper we show that dimensionality reduction (i.e., JohnsonLindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between lowdimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces ..."
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Cited by 14 (3 self)
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In this paper we show that dimensionality reduction (i.e., JohnsonLindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between lowdimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces. We also show that surfaces with bounded doubling dimension can be embedded into low dimension with small additive error. Finally, we show that for points with polynomial motion, the radius of the smallest enclosing ball can be preserved under dimensionality reduction. 1
Embedding metric spaces in their intrinsic dimension
 In SODA’08
, 2007
"... A fundamental question of metric embedding is whether the metric dimension of a metric space is related to its intrinsic dimension. That is whether the dimension in which it can be embedded in some real normed space is implied by the intrinsic dimension which is reflected by the inherent geometry of ..."
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Cited by 12 (7 self)
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A fundamental question of metric embedding is whether the metric dimension of a metric space is related to its intrinsic dimension. That is whether the dimension in which it can be embedded in some real normed space is implied by the intrinsic dimension which is reflected by the inherent geometry of the space. The existence of such an embedding was conjectured by Assouad and was later posed as an open problem by others. This question is tightly related to a major goal of many practical application fields: developing tools to represent intrinsically low dimensional metric data sets in a succinct manner. In this paper we give the first algorithmic technique with formal guarantees for finding faithful and low dimensional representations of data lying in high dimensional space. Our main theorem states that every finite metric space X embeds into Euclidean space with dimension O(dim(X)/ɛ) and distortion O(log 1+ε n), where dim(X) is the doubling dimension of the space X. Moreover, we show that X can be embedded into dimension Õ(dim(X)) with constant average distortion and ℓqdistortion for any q < ∞. Our technique also provides a dimensiondistortion tradeoff and an extension of Assouad’s theorem, providing distance oracles that improve known construction when dim(X) = o(log X). 1
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion, 2006. Arxiv
"... This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling dist ..."
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Cited by 10 (4 self)
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This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of ǫ, with the guarantee that for each ǫ the distortion of a fraction 1−ǫ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓqdistortions are small. Specifically, our embeddings have constant average distortion and O ( √ log n) ℓ2distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O ( √ 1/ǫ). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O ( √ 1/ǫ). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of Õ(log 2 (1/ǫ)), which implies constant ℓqdistortion for every fixed q < ∞. 1
Local embeddings of metric spaces
 PROCEEDINGS OF THE 39TH ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2007
"... In many application areas, complex data sets are often represented by some metric space and metric embedding is used to provide a more structured representation of the data. In many of these applications much greater emphasis is put on the preserving the local structure of the original space than on ..."
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Cited by 10 (3 self)
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In many application areas, complex data sets are often represented by some metric space and metric embedding is used to provide a more structured representation of the data. In many of these applications much greater emphasis is put on the preserving the local structure of the original space than on maintaining its complete structure. This is also the case in some networking applications where “small world” phenomena in communication patterns has been observed. Practical study of embedding has indeed involved with finding embeddings with this property. In this paper we initiate the study of local embeddings of metric spaces and provide embeddings with distortion depending solely on the local structure of the space.
Spanners with slack
 Proceedings of the 14th European symposium on algorithms
, 2006
"... Abstract. Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily l ..."
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Cited by 7 (1 self)
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Abstract. Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distances, but are then required to incur only a constant (independent of n) distortion on the remaining distances? We answer this question in the affirmative, thus complementing similar recent results on embeddings with slack into ℓp spaces. For instance, we show that if we ignore an ɛ fraction of the distances, we can get spanners with O(n) edgesand O(log 1) distortion for the remaining distances. ɛ We also show how to obtain sparse and lowweight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces. 1
Compact Routing with Slack in Low Doubling Dimension ABSTRACT
"... We consider the problem of compact routing with slack in networks of low doubling dimension. Namely, we seek nameindependent routing schemes with (1 + ɛ) stretch and polylogarithmic storage at each node: since existing lower bound precludes such a scheme, we relax our guarantees to allow for (i) a s ..."
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Cited by 6 (1 self)
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We consider the problem of compact routing with slack in networks of low doubling dimension. Namely, we seek nameindependent routing schemes with (1 + ɛ) stretch and polylogarithmic storage at each node: since existing lower bound precludes such a scheme, we relax our guarantees to allow for (i) a small fraction of nodes to have large storage, say size of O(n log n) bits, or (ii) a small fraction of sourcedestination pairs to have larger, but still constant, stretch. In this paper, given any constant ɛ ∈ (0, 1), any δ ∈ Θ(1 / polylog n) and any connected edgeweighted undirected graph G with doubling dimension α ∈ O(log log n) andarbitrary node names, we present
Additive Spanners and (α, β)Spanners
"... An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). How ..."
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Cited by 6 (2 self)
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An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)spanners. Our first result is an additive (1, 6)spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively wellapproximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparsenessdistortion tradeoffs. Our second result addresses the problem of which (α, β)spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
A Nashtype Dimensionality Reduction for Discrete Subsets of L2
"... Pursuing a line of work begun by Whitney, Nash showed that every C 1 manifold of dimension d can be embedded in R 2d+2 in such a manner that the local structure at each point is preserved isometrically. We provide an analog of this result for discrete subsets of Euclidean space. For perfect preserva ..."
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Cited by 3 (1 self)
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Pursuing a line of work begun by Whitney, Nash showed that every C 1 manifold of dimension d can be embedded in R 2d+2 in such a manner that the local structure at each point is preserved isometrically. We provide an analog of this result for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute nearisometric embedding of neighborhoods of bounded cardinality. There are a variety of empirical situations, abstracted to metric space tasks, in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widelyused heuristics such as Isomap and Locally Linear Embedding. In such situations we offer the possibility of dimension reductions unobtainable with global methods, because the dimension of our locally (1 + ɛ)distorting embedding is proportional to ɛ −2 log k, where k is the cardinality of the neighborhoods where distances are preserved and is independent of the number of points in the metric space. One may view our work as a local version of the widelyused JohnsonLindenstrauss lemma. We use a device of Nash, together with more recent metric embedding methods, to compose local dimensionality reducing embeddings within a global immersion (that preserves short distances), and with mild additional assumptions, a global embedding that also keeps distant points wellseparated. We provide an efficient randomized algorithm for the embeddings.