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232 embedding the embedding is a contraction let x y

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Unformatted text preview: in different connected components Ci , Cj d(x, S ), d(y, S ) ≥ c1 ∆ with constant probability Now, for ri , rj random numbers chosen uniformly from [1, 2] |ri d(x, S ) − rj d(y, S )| ≥ c1 ∆ with constant probability CSE 254 - Metric Embeddings - Winter 2007 – p. 22/2 Embedding We will now define the embedding: For each ∆ ∈ {2j |1 ≤ 2j ≤ Diam(G)} perform 4 log n random decompositions For each component Ck in a decomposition uniformly pick a random rk from [1, 2] For x ∈ Ck define its coordinate as rk · d(x, S ) This defines a mapping f∆,i : x → rk · d(x, S ) for all ∆ and all i ∈ {2j |1 ≤ 2j ≤ log n} Finally, let f :x→ „ 1 f∆,i (x) : ∆, i 2 log n « CSE 254 - Metric Embeddings - Winter 2007 – p. 23/2 Embedding The embedding is a contraction Let x, y in V (G), then f (x) − f (y ) 2 ∆,i ≤ ≤ = 1 (f∆,i (x) − f∆,i (y ))2 (2 log n)2 1 4 log2 n (2d(x, y ))2 ∆,i 1 4 log2 n(4d(x, y )2 ) = d(x, y )2 4 log2 n CSE 254 - Metric Embeddings - Winter 2007 – p. 24/2 Embedding √ The embedding has distortion O( log n) Let x, y in V (G), and pick a ∆ such that 34∆ < d(x, y ) < 68∆ then f (x) − f (y ) ≥ 2 ≥ i ≥ i 1 (f∆,i (x) − f∆,i (y ))2 (2 log n)2 1 (Ω(1)d(x, y ))2 (2 log n)2 1 (d(x, y ))2 Ω(1) log n CSE 254 - Metric Embeddings - Winter 2007 – p. 25/2 Applications Using this result, we can obtain √ a O( log n)-approximative max flow min cut theorem for multicommodity flow problems in planar graphs CSE 254 - Metric Embeddings - Winter 2007 – p. 26/2 Further results Definition. For a set of k points S in RL the volume Evol(S ) is the k − 1 dimensional ℓ2 - volume of the convex hull of S Definition. The volume of a k-point metric space (S,d) is V ol(S ) = sup Evol(f (S )) f :S →ℓ2 (the maximum being taken over all contractions f ) CSE 254 - Metric Embeddings - Winter 2007 – p. 27/2 Further results Definition. A (k,c)-volume preserving embedding of a metric space (S,d) is a contraction f : X → ℓ2 where for all P ⊂ S with |P | = k , (1) V ol(S ) Evol(f (S ) 1/(k−1) ≤c The k-distortion of f is (2) sup P ⊆S,|P |=k V ol(S ) Evol(f (S ) 1/(k−1) With the help of some results from Feige, we can prove the following: Theorem. Rao’s Theorem For every finite planar metric of cardinality n there exists a √ (k,c)-volume preserving embedding of k-distortion O ( log n) CSE 254 - Metric Embeddings - Winter 2007 – p. 28/2 References S. Rao. Small distortion and volume preserving embeddings for planar and euclidean metrics. In Proceedings of the 15th Annual Symposium on Computational Geometry, ACM Press, 1999 P. Klein, S. Rao and S. Plotkin. Excluded minors, network decompositions, and multicommodity flow. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993 U. Feige. Approximating the bandwidth via volume respecting embeddings. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998 J. Matousek. Lectures on Discrete Geometry. Springer-Verlag, New York, 2002 CSE 254 - Metric Embeddings - Winter 2007 – p. 29/2...
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