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Lecture 5 Notes

# 182 red nodes blue nodes claim av2 is disjoint from

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Unformatted text preview: ∆ apart from A(v1 )) Question is: could a vertex x of A(v1 ) belong to one of t′′ h′ , t′′ h′ , t′′ h′ , say t′′ h′ ? 112222 11 Then, d(x, h′ ) 1 ≤ 4∆, thus d(x, u) ≤ 8∆ Without loss of generality, let’s say x is on path v1 h′ 2 Now d(x, h′ ) 2 ≤ 5∆ − 1 (because h′ and h′ are within consecutive ∆ − 1 levels) 1 2 So, d(x, v ) ≤ 9∆ − 1 and thus d(u, v ) ≤ 17∆ − 1, a contradiction CSE 254 - Metric Embeddings - Winter 2007 – p. 18/2 Red nodes, blue nodes Claim. A(v2 ) is disjoint from blue nodes A(u), A(v ), A(w) Proof. Repeating the arguments for A(v1 ), we only need to worry about paths t′′ h′ , t′′ h′ , t′′ h′ intersecting A(v2 ) 112222 Pick an x in T2 on t′′ h′ and y in A(v2 ) 11 Without loss of generality, let’s say y is on path v2 h2 Then, restricting our distance metric to T2 we get: d(v2 , x) ≥ d(v2 , h′ ) − (∆ − 1) (consecutive ∆ − 1 levels) 1 So, d(v2 , x) ≥ d(v2 , h1 ) − 2∆ − (∆ − 1) = d(v2 , h1 ) − 3∆ + 1 But then d(v2 , x) ≥ d(v2 , h2 ) − (∆ − 1) − 3∆ + 1 = d(v2 , h2 ) − 4∆ + 2 But, y being in A(v2 ), d(y, h2 ) ≥ 4∆, so d(v2 , h2 ) − 4∆ + 2 ≥ d(v2 , y ) + 2 Thus, d(v2 , x) ≥ d(v2 , y ) + 2, x and y are therefore distinct CSE 254 - Metric Embeddings - Winter 2007 – p. 19/2 Red nodes, blue nodes Claim. A(v3 ) is disjoint from blue nodes A(u), A(v ), A(w) Proof. The exact same technique as in the previous proof covers all the cases we need to consider ... CSE 254 - Metric Embeddings - Winter 2007 – p. 20/2 Property 1 - End of Proofsketch By contracting the super nodes, we observe a K3,3 This violates the assumption of the graph being planar (Kuratowski) ∴ For each component C , we have Diam(C ) < 34∆ By induction, Klein, Plotkin and Rao in their paper actually proof the following stronger statement: Theorem. If G excludes Kr,r as a minor, any connected component obtained through r iterations of the described decomposition method has diameter O(r3 ∆) CSE 254 - Metric Embeddings - Winter 2007 – p. 21/2 Properties Given a random decomposition, with parameter ∆ Each component in the decomposition has diameter at most O(∆) For each x ∈ V (G) we have P[d(x, S ) ≥ c1 ∆] ≥ c2 We now furthermore have: For any x, y ∈ V (G), with d(x, y ) ≥ 34∆ x, y ∈ S with constant probability / x, y are...
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