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Ordinal embedding: Approximation algorithms and dimensionality reduction
, Article 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008, Boston, MA, 25 August 2008 through 27 August 2008 ; Volume 5171 LNCS , 2008 , Pages 21-34 ; 03029743 (ISSN) ; 9783540853626 (ISBN) ; Demaine, E. D ; Hajiaghayi, M ; Sidiropoulos, A ; Zadimoghaddam, M ; Sharif University of Technology
2008
Abstract
This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target...
Plane embeddings of planar graph metrics
, Article Discrete and Computational Geometry ; Volume 38, Issue 3 , 2007 , Pages 615-637 ; 01795376 (ISSN) ; Demaine, E. D ; Hajiaghayi, M ; Moharrami, M ; Sharif University of Technology
Springer New York
2007
Abstract
Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ(√n) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by...
Plane embeddings of planar graph metrics
, Article 22nd Annual Symposium on Computational Geometry 2006, SCG'06, Sedona, AZ, 5 June 2006 through 7 June 2006 ; Volume 2006 , 2006 , Pages 197-206 ; 1595933409 (ISBN); 9781595933409 (ISBN) ; Demaine, E. D ; Hajiaghayi, M ; Moharrami, M ; ACM SIGACT; ACM SIGGRAPH ; Sharif University of Technology
Association for Computing Machinery
2006
Abstract
Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ(√n) distortion in the worst case [19, 1], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by...