Proceedings ArticleDOI
The geometry of graphs and some of its algorithmic applications
Nathan Linial,Eran London,Yuri Rabinovich +2 more
- pp 577-591
Reads0
Chats0
TLDR
Some implications of viewing graphs as geometric objects are explored and efficient algorithms for embedding graphs low-dimensionally with a small distortion are developed.Abstract:
We explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. We develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. >read more
Citations
More filters
Proceedings Article
Similarity Search in High Dimensions via Hashing
TL;DR: Experimental results indicate that the novel scheme for approximate similarity search based on hashing scales well even for a relatively large number of dimensions, and provides experimental evidence that the method gives improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition.
Journal ArticleDOI
Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions
Alexandr Andoni,Piotr Indyk +1 more
TL;DR: An algorithm for the c-approximate nearest neighbor problem in a d-dimensional Euclidean space, achieving query time of O(dn 1c2/+o(1)) and space O(DN + n1+1c2 + o(1) + 1/c2), which almost matches the lower bound for hashing-based algorithm recently obtained.
Proceedings ArticleDOI
Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions
Alexandr Andoni,Piotr Indyk +1 more
TL;DR: An algorithm for the c-approximate nearest neighbor problem in a d-dimensional Euclidean space, achieving query time of O and space O almost matches the lower bound for hashing-based algorithm recently obtained in [27].
Journal ArticleDOI
The geometry of graphs and some of its algorithmic applications
TL;DR: Efficient algorithms for embedding graphs low-dimensionally with a small distortion, and a new deterministic polynomial-time algorithm that finds a (nearly tight) cut meeting this bound.
Proceedings ArticleDOI
Probabilistic approximation of metric spaces and its algorithmic applications
TL;DR: It is proved that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion.
References
More filters
Book
Pattern classification and scene analysis
Richard O. Duda,Peter E. Hart +1 more
TL;DR: In this article, a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition is provided, including Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
Journal ArticleDOI
On the Shannon capacity of a graph
TL;DR: It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.
Book
A course in combinatorics
J.H. van Lint,Richard M. Wilson +1 more
TL;DR: The second edition of a popular book on combinatorics as discussed by the authors is a comprehensive guide to the whole of the subject, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes.
Book
The volume of convex bodies and Banach space geometry
TL;DR: In this paper, the authors present a proof of the QS theorem for weak Hilbert spaces and weak cotype for weak type 2... and weak Hilbert space for weak Cotype.