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Graph Powering Cont.

Graph Powering Cont. PCP proof by Irit Dinur Presentation by: Alon Vekker. Last lecture G’ construction. V’ = V B=C · t C = const. E’ = For every two vertices at distance at most t we have a new edge between them. From last lecture. V. V’ = V. Σ. C(u,v). gap’ ≥ t/O(1)*min(gap,1/t).

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Graph Powering Cont.

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  1. Graph Powering Cont. PCP proof by Irit Dinur Presentation by: Alon Vekker

  2. Last lecture G’ construction • V’ = V • B=C·t C = const. • E’ = For every two vertices at distance at most t we have a new edge between them.

  3. From last lecture V V’ = V Σ C(u,v) gap’ ≥ t/O(1)*min(gap,1/t) gap

  4. We built the graph linearly. • We look at vertices at distance at most t. • We look at opinions at distance at most B.

  5. Plurality assignment • Definition: : V’ is defined as follows: the opinion of w about v. • Definition: : V is defined as follows: (v) is the plurality of opinions of about v. • Plurality : al least • Unweighted edges!!!

  6. Example for σ’ We use here : t ≤ 2 Σ = {1,2,3} a

  7. σ(a): Flat plurality a

  8. Last week Analysis • Definition: F is a subset of E which includes all edges that are not satisfied by σ. • |F|/|E|≥gap • We throw edges from F until |F|/|E|=min(gap,1/t)=:

  9. Last week Analysis [e’ passes through F] [e’ completely misses F] ( by the lemma ) ( since for )

  10. Example Too long a e’ 2 a b 1 v u F

  11. Another look Is it working? Un weighted plurality

  12. E’: What weight to give to an edge? • Pick a random vertex a • Take a step along a random edge out of the current vertex. • Decide to stop with probability 1/t. • Stop if you passed B steps already.

  13. Example • A is the plurality but they are too far. B a a b a a u b a b a a b v b b a a b a a

  14. Why do we get weighted edges? 2 3 2 b 3 2 2 a 1 2 3 3 1 3 2 1 1 b 1 1 1 1

  15. Edge Weight: • (a,b) G’ • Dist(a,b) ≤ t • The weight on the adge (a,b) is:

  16. New plurality To define (v): consider the probability distribution on vertices as follows: • Do SW starting from v, ending on w.

  17. Lemma 1: • if a path a b in G uses an edge (u,v) • Then, if: • (u,v) F THEN : σ’ violates the constraint on edge e’. That leads us to a conclusion… • When the length of the path < B

  18. Lemma 2: • Let G be an (n,d,λ)-expander and F subset of E. Then the probability that a random walk, starting in the zero-th step from a random edge in F, passes through F on its t step is bounded by • Later used to prove PCP theorem.

  19. Final Analysis [e’ completely misses F] Lemma 1 Lemma 2

  20. Proof of lemma 1: • Suppose we don’t stop SW after B steps • Our Σ will depend on the number of vertices. • Its to big so we must stop after B steps.

  21. calculations • Lets count the probability of a path longer then B: • And therefore we get:

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