Decomposing graphs into regions of small diameter
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Approximate max-flow min-(multi)cut theorems and their applications
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
A polynomial time approximation scheme for minimum routing cost spanning trees
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
New approximation techniques for some ordering problems
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Divide-and-conquer approximation algorithms via spreading metrics
Journal of the ACM (JACM)
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Approximating min-sum k-clustering in metric spaces
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
IDMaps: a global internet host distance estimation service
IEEE/ACM Transactions on Networking (TON)
Lectures on Discrete Geometry
An improved approximation algorithm for the 0-extension problem
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On average distortion of embedding metrics into the line and into L1
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Constructing internet coordinate system based on delay measurement
Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement
Practical, distributed network coordinates
ACM SIGCOMM Computer Communication Review
PIC: Practical Internet Coordinates for Distance Estimation
ICDCS '04 Proceedings of the 24th International Conference on Distributed Computing Systems (ICDCS'04)
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Measured Descent: A New Embedding Method for Finite Metrics
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Triangulation and Embedding Using Small Sets of Beacons
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Journal of the ACM (JACM)
Big-bang simulation for embedding network distances in Euclidean space
IEEE/ACM Transactions on Networking (TON)
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Metric Embeddings with Relaxed Guarantees
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Ramsey-type theorems for metric spaces with applications to online problems
Journal of Computer and System Sciences - Special issue on FOCS 2001
On space-stretch trade-offs: lower bounds
Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures
Embeddings of surfaces, curves, and moving points in euclidean space
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Local embeddings of metric spaces
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Compact routing with slack in low doubling dimension
Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing
Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Ultra-low-dimensional embeddings for doubling metrics
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Embedding metric spaces in their intrinsic dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Graph Augmentation via Metric Embedding
OPODIS '08 Proceedings of the 12th International Conference on Principles of Distributed Systems
On low dimensional local embeddings
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Distributed similarity search in high dimensions using locality sensitive hashing
Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology
Triangulation and embedding using small sets of beacons
Journal of the ACM (JACM)
Streaming Embeddings with Slack
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Ultra-low-dimensional embeddings for doubling metrics
Journal of the ACM (JACM)
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Volume in general metric spaces
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
Low distortion metric embedding into constant dimension
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Topic segmentation: application of mathematical morphology to textual data
ISMM'11 Proceedings of the 10th international conference on Mathematical morphology and its applications to image and signal processing
Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Using petal-decompositions to build a low stretch spanning tree
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few.The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications.A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion.Bourgain's result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion.Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain's theorem.In fact, our embedding possesses a much stronger property. We define the lq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible lq-distortion for all 1 ≤ q ≤ ∞ simultaneously.These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension.The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain's embedding.We prove that any metric space on n points embeds into Lp with distortion O(log n) in dimension O(log n). This provides an optimal bound on the dimension of the embedding.Somewhat surprisingly, we show that a further small improvement is possible at a small price in the distortion, obtaining an embedding with distortion O(log1+θ n) in optimal dimension O(θ-1 log n/log log n), for any θ 0. It is worth noting that with the small loss in the distortion this improves upon the best known embedding of arbitrary spaces into Euclidean space, where dimension reduction is used.Our techniques also allow to obtain the optimal distortion for embedding into Lp with nearly tight dimension. For any 1 ≤ p ≤ ⊂ and any 1 ≤ k ≤ p, we give an embedding into Lp with distortion O(⌈ log n/k ⌉) in dimension 2O(k)log n.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results. These metric decompositions seem of independent interest.