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1 Graph Powering Cont. PCP proof by Irit Dinur Presented by Israel Gerbi
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2 Goal Reduction Goal: Input: Constraint graph (G=(V,E),C, ) where G is an (n,d, ) expander, and < d, Output: A new graph (G’,C’) with larger gap (denoted gap’), where If gap = 0, Otherwise.
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3 Last Lecture G’ Construction V’ = V B=C·t C = const.
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4 E’: How to generate 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. Throw edge if above path has length>B We get weighted edges, why?
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5 C’: New constraints
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6 Plurality Assignment
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7 New Plurality: Formal Definition
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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)
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9 Gap’ Analysis Reminder Lemma from last lesson Over all paths from a to b (weighted e’ edges)
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10 To Work… Starting with more Definitions S := Total number of steps in our RW N F := Number of steps that were in F N F * := Number of steps that were in F, if our RW wasn’t limited to B steps
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11 Motivation In more detail, we show:
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12 Motivation cont. Second Moment Method says: We wanted to show:
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13 Back to the Beginning We can now choose t so the new gap would be twice as large!
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14 Expectation of N F The graph is d regular
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15 Cutting Off the Tail We will now bound
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16 The Tail
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17 (1) Proof Combining the two results above we get: We finished proving (1). We now turn to (2)
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18 (2) Proof We now show(2): Obviously,
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19 (2) Proof Cont. Lemma from first lesson
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20 (2) Proof Cont.
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21 Second Moment Method Lemma (Second Moment Method): If X is a nonnegative r.v then Proof: Cauchy Schwartz inequality
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22 Second Moment Method Proof We have: Arranging: Therefore:
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