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 kpoint 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 kdistortion 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 kdistortion 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. SpringerVerlag, New York, 2002 CSE 254  Metric Embeddings  Winter 2007 – p. 29/2...
View
Full Document
 Winter '07
 Algorithms, Graph Theory, Metric space, Planar graph, Satish Rao, metric embeddings

Click to edit the document details