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Bounded geometries, fractals, and lowdistortion embeddings
"... The doubling constant of a metric space (X; d) is thesmallest value * such that every ball in X can be covered by * balls of half the radius. The doubling dimension of X isthen defined as dim(X) = log2 *. A metric (or sequence ofmetrics) is called doubling precisely when its doubling dimension is ..."
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Cited by 211 (42 self)
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is bounded. This is a robust class of metric spaceswhich contains many families of metrics that occur in applied settings.We give tight bounds for embedding doubling metrics into (lowdimensional) normed spaces. We consider bothgeneral doubling metrics, as well as more restricted families such as those
LowDistortion Embeddings of Trees
 Journal of Graph Algorithms and Applications
, 2003
"... We prove that every tree T = (V,E) on n vertices with edges of unit length can be embedded in the plane with distortion O( n); that is, we construct a mapping f : V > R² such that #(u, v) f(v) O( n) #(u, v) for every u, v V ,where#(u, v) denotes the length of the path from u to v in T ..."
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Cited by 10 (1 self)
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We prove that every tree T = (V,E) on n vertices with edges of unit length can be embedded in the plane with distortion O( n); that is, we construct a mapping f : V > R² such that #(u, v) f(v) O( n) #(u, v) for every u, v V ,where#(u, v) denotes the length of the path from u to v
Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. Technical Report 2003/235, Cryptology ePrint archive, http://eprint.iacr.org, 2006. Previous version appeared at EUROCRYPT 2004
 34 [DRS07] [DS05] [EHMS00] [FJ01] Yevgeniy Dodis, Leonid Reyzin, and Adam
, 2004
"... We provide formal definitions and efficient secure techniques for • turning noisy information into keys usable for any cryptographic application, and, in particular, • reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying mater ..."
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Cited by 532 (38 self)
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We provide formal definitions and efficient secure techniques for • turning noisy information into keys usable for any cryptographic application, and, in particular, • reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor reliably extracts nearly uniform randomness R from its input; the extraction is errortolerant in the sense that R will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, R can be used as a key in a cryptographic application. A secure sketch produces public information about its input w that does not reveal w, and yet allows exact recovery of w given another value that is close to w. Thus, it can be used to reliably reproduce errorprone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.
Lowdistortion Embeddings of General Metrics Into the
"... A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absol ..."
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A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results
LowDistortion Embeddings of Finite Metric Spaces
 in Handbook of Discrete and Computational Geometry
, 2004
"... INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their diss ..."
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Cited by 65 (2 self)
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INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is dicult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way. For example, it would be very nice if we could assign to each x 2 X a point f(x) in the plane in such a way that D(x; y) equals the Euclidean distance of f(x) and f(y). Such a representation would allow us to see the structure of the metric space: tight clusters, isolated points, and so on. Another advantage would be that the metric would now be represented by only 2n real numbers, the coordinates of the n points in the plane, instead of numbers as before. Moreover, many quantities concern
Multiple Description Coding: Compression Meets the Network
, 2001
"... This article focuses on the compressed representations of the pictures ..."
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Cited by 435 (9 self)
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This article focuses on the compressed representations of the pictures
Lowdistortion embeddings of general metrics into the line
 In STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absol ..."
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Cited by 26 (8 self)
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A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results
Probabilistic Approximation of Metric Spaces and its Algorithmic Applications
 In 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized ..."
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Cited by 361 (33 self)
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The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized
The complexity of lowdistortion embeddings between point sets
 Proceedings of SODA 2005
, 2005
"... Abstract We prove that it is NPhard to approximate by a ratio better than 3 the minimum distortionof a bijection between two given finite threedimensional sets of points. ..."
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Cited by 17 (1 self)
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Abstract We prove that it is NPhard to approximate by a ratio better than 3 the minimum distortionof a bijection between two given finite threedimensional sets of points.
Approximation Algorithms for LowDistortion Embeddings Into LowDimensional Spaces
 in Proceedings of the 16th Annual ACMSIAM Symposium on Discrete Algorithms
, 2005
"... Abstract We present several approximation algorithms for theproblem of embedding metric spaces into a line, and into the twodimensional plane. Among other results, wegive an O(pn)approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, t ..."
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Cited by 28 (8 self)
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Abstract We present several approximation algorithms for theproblem of embedding metric spaces into a line, and into the twodimensional plane. Among other results, wegive an O(pn)approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph
Results 1  10
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