Lecture 12: Constant Degree Lossless Expanders
نویسندگان
چکیده
In this lecture, we will construct constant degree D vertex-expanders with expansion of (1 − ε)D (a.k.a. constant degree lossless expanders). This beautiful construction is due to Capalbo-ReingoldVadhan-Wigderson. Concretely, for every constant ε > 0 and every N , we will construct a bipartite graph (L,R,E), |L| = N , |R| = M = poly(ε)N), with left degree D = poly(1ε ), such that every subset S of L of size at most poly(ε)M D , the size of its neighborhood Γ(S) is at least (1− ε) ·D · |S|. In contrast, our earlier lossless condenser construction (which actually is a lossless expander) had polylogarithmic degree.
منابع مشابه
Randomness Conductors and Constant-Degree Lossless Expanders [Extended Abstract]
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1− ǫ)D, where D is the degree and ǫ is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the...
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In the last lecture we described the construction of the zigzag product of graphs and completed the analysis of its expansion. This was the final building block for our construction of arbitrarily large d-regular expanders, for a constant d. In particular, we had d = 372 and achieved λ2 ≤ λ < d/2. Notice that, if we needed a larger expansion, we could always recur to graph powering. Given G, wi...
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