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Unformatted text preview: Inapproximability for planar embedding problems Jeff Edmonds 1 Anastasios Sidiropoulos 2 Anastasios Zouzias 3 1 York University 2 TTIChicago 3 University of Toronto January, 2010 A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 1 / 21 Metric Space M = ( X , D ) is a metric space. X is a set. D is a distance function on X , i.e., satisfies triangle inequality. Examples Any normed space. Graphs with shortest path distance. ... ... ... A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 2 / 21 Embedding between metric spaces Given M = ( X , d X ) and M = ( Y , d Y ) . Embedding f : X mapsto Y . Metric distortion f has distortion if x 1 , x 2 X d X ( x 1 , x 2 ) d Y ( f ( x 1 ) , f ( x 2 )) d X ( x 1 , x 2 ) . dist ( f ) = maxexp( f ) maxcontr( f ) Wellstudied subject: Worst case distortion. Relative Embeddings: Given X , find nearoptimal embedding f : X mapsto Y efficiently. A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 3 / 21 Embedding between metric spaces Given M = ( X , d X ) and M = ( Y , d Y ) . Embedding f : X mapsto Y . Metric distortion f has distortion if x 1 , x 2 X d X ( x 1 , x 2 ) d Y ( f ( x 1 ) , f ( x 2 )) d X ( x 1 , x 2 ) . dist ( f ) = maxexp( f ) maxcontr( f ) Wellstudied subject: Worst case distortion. Relative Embeddings: Given X , find nearoptimal embedding f : X mapsto Y efficiently. A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 3 / 21 Computational problems (Approximate min. distortion) Bijection and Injection Bijection Given two finite metric spaces X , Y of the same size n . Approximate the minimum distortion bijection f : X mapsto Y . Introduced in [KRS04]. Injection Given finite metric space X of size n and infinite metric space Y with fixed dimensionality. Approximate the minimum distortion injection of f : X mapsto Y . Remark: Although different problems, share the same approximability. Notation vs. : Given X it is NPhard to check if f : X mapsto Y with distortion or every f has distortion > . Notice that 1 < . A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 4 / 21 Computational problems (Approximate min. distortion) Bijection and Injection Bijection Given two finite metric spaces X , Y of the same size n . Approximate the minimum distortion bijection f : X mapsto Y . Introduced in [KRS04]. Injection Given finite metric space X of size n and infinite metric space Y with fixed dimensionality. Approximate the minimum distortion injection of f : X mapsto Y ....
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This note was uploaded on 02/13/2012 for the course CSE 4111 taught by Professor Edmonds during the Winter '12 term at York University.
 Winter '12
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