Lecture 9 : AC 0 Circuit Size of SUB ( G ) , Continued
نویسنده
چکیده
Throughout today’s lecture, we fix a connected graph G and a strict threshold weighting θ : E(G)→ [0, 2] (such that ∆θ(H) := |V (H)| − ∑ e∈E(H) θ(e) > 0 for all ∅ ⊂ H ⊂ G). Also throughout today’s lecture, let α be a uniform random element of [n]V (G) (independent of Xθ). We think of α as a random choice of vertices from each color class of G ↑n. For each subgraph H ⊆ G, we have the isomorphic subgraph H(α) ⊆ G↑n given by E(H) := {{v, w} ∈ E(G↑n) : {v, w} ∈ E(H) and i = αv and j = αw}.
منابع مشابه
Lecture 8 : AC 0 Circuit Size of SUB ( G )
We say θ is strict if ∆θ(H) > 0 for all ∅ ⊂ H ⊂ G. • Xθ is the random subgraph of G↑n where, independently for each {v(i), w(j)} ∈ E(G↑n), P[ {v, w} ∈ E(Xθ) ] = n−θ({v,w}). We have E[ subH(Xθ) ] = n ∆θ(H). If θ is strict, then subG(Xθ) is asymptotically Poisson(1) and subH(Xθ) is highly concentration around its mean for all ∅ ⊂ H ⊂ G. Convention. In the context of lower bounds for SUB(G), the a...
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