Lecture 5 Notes

132 blue super nodes let us dene the following blue

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Unformatted text preview: on of three paths of the tree T3 from the root v3 to vertices t1 , t2 , t3 (but not including t1 , t2 , t3 ) each of t1 , t2 , t3 is at distance 4∆ from one of u, v, w Similarly, define A(v2 ), A(v1 ) with respect to the t′ ’s and t′′ ’s i i CSE 254 - Metric Embeddings - Winter 2007 – p. 13/2 Blue super nodes Let us define the following blue super nodes: A(u), A(v ), A(w) A(u) is the union of the path on tree T3 joining u and t1 the path on tree T2 joining h1 and t′ 1 the path on tree T1 joining h′ and t′′ 1 1 A(v ) and A(w) are defined similarly CSE 254 - Metric Embeddings - Winter 2007 – p. 14/2 Super nodes v2 h’3 v1 h’2 111111111111 000000000000 111111111111 000000000000 v 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 h’1 111111111111 000000000000 C 111111111111 000000000000 111111111111 000000000000 w 111111111111 000000000000 111111111111 000000000000 h3 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 u 111111111111 000000000000 h1 111111111111 000000000000 h2 v3 CSE 254 - Metric Embeddings - Winter 2007 – p. 15/2 Super nodes are disjoint Claim. A(u), A(v ), A(w) (Blue super nodes) are pairwise disjoint Proof. Each blue node is only 8∆ in diameter (see diagram) Yet, they each contain one of u, v, w , any two of which are > Claim. 16∆ apart A(v1 ), A(v2 ), A(v3 ) (Red super nodes) are pairwise disjoint A(v1 ), A(v2 ), A(v3 ) are separated by the decomposition Each of h1 , h2 , h3 is > 4∆ away from A(v2 ) Thus A(v2 ) ∩ T3 = ∅ and a fortiori A(v2 ) ∩ A(v3 ) = ∅ Same argument applies to A(v1 ) with respect to either of A(v2 ) and A(v3 ) Proof. Finally, similar arguments will show that any red super node is disjoint from any blue super node CSE 254 - Metric Embeddings - Winter 2007 – p. 16/2 Super nodes 1111111111 0000000000 1111111111 0000000000 v2 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111111111111111111 0000000000000000000000000 1111111111 t’1 0000000000 t’3 t’2 1111111111111111111111111 0000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111111...
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