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Graph Theory Presentation by Irit Dinur: An In-Depth Analysis

Explore the concepts of graph powering, PCP proof, and G' construction in this detailed presentation. Learn about plurality assignment, unweighted edges, and weighted edges. Delve into the analysis of last week's examples and understand the importance of edge weight in graph theory.

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Graph Theory Presentation by Irit Dinur: An In-Depth Analysis

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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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