Lecture 5 Notes

92 property 1 proofsketch now let v2 be the root of

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Unformatted text preview: 00000000000 2 111111111111111111111111 000000000000000000000000 C h2 2 t2 t3 T3 v3 CSE 254 - Metric Embeddings - Winter 2007 – p. 9/2 Property 1 - Proofsketch Now, let v2 be the root of the BF S -tree of the previous level ′ ′ ′ It has disjoint paths h1 t1 , h2 t2 , h3 t3 of length 4∆ and if we ′ ′ ′ let h1 ,h2 ,h3 be their midpoints ′ ′ d(hi , hj ) > 8∆ for all i = j Similarly, for the first BF S -tree of the decomposition Look at disjoint paths in the tree ′′ ′′ ′′ Define h1 ,h2 ,h3 as before ′′ ′′ ′′ h1 ,h2 ,h3 are pairwise more than 4∆ apart CSE 254 - Metric Embeddings - Winter 2007 – p. 10/2 Property 1 - Diagram 2 v2 T2 t’1 t’3 h’3 t’2 h’2 2 111111111111 000000000000 111111111111 000000000000 2 v 111111111111 000000000000 2 111111111111 000000000000 h2 111111111111 000000000000 111111111111 000000000000 h’1 111111111111 000000000000 C 111111111111 000000000000 >12 111111111111 000000000000 w 111111111111 000000000000 2 111111111111 000000000000 >12 h3 2 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 >12 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 u 2 h1 111111111111 000000000000 111111111111 000000000000 v3 CSE 254 - Metric Embeddings - Winter 2007 – p. 11/2 Property 1 - Diagram 3 v2 h’’3 2 t’’3 h’3 >8 v1 t’’2 h’’2 2 T1 t’’1 2 h’’1 h’2 >8 111111111111 000000000000 111111111111 000000000000 v 111111111111 000000000000 >8 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. 12/2 Red super nodes Definition. A super node of a graph G is a connected subgraph Let us define the following red super nodes: A(v1 ), A(v2 ), A(v3 ) A(v3 ) is the uni...
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