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Lecture 5 Notes

# 72 property 1 proofsketch suppose there is a

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Unformatted text preview: diameter of each component therefore has to be bounded CSE 254 - Metric Embeddings - Winter 2007 – p. 7/2 Property 1 - Proofsketch Suppose there is a component C containing u, v such that d(u, v ) ≥ 34∆ Let w be the midpoint of the path between them (within C ) d(u, w), d(w, v ) ≥ 17∆ Let v3 be the root of the last BF S -tree used to obtain the component ∃ disjoint paths ut1 , wt2 , vt3 of length 4∆ in the tree Let h1 ,h2 ,h3 be their midpoints, i.e. d(u, h1 ) = 2∆, etc We then have that d(hi , hj ) > 12∆ for all i = j CSE 254 - Metric Embeddings - Winter 2007 – p. 8/2 Property 1 - Diagram 1 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 v 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 2 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 17 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 w 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 h3 111111111111111111111111 000000000000000000000000 2 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 2 111111111111111111111111 000000000000000000000000 17 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 u h1 2 t1 111111111111111111111111 0000000000000...
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