2 Spectral Expansion . . . 3 Random Walks and Bounding . . . 4 L vs. RL and Zig zag expanders Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 2 / 8
O(n) edges. Almost everywhere Byzantine Agreement: loose unamity to solve bounded degree network case. Random bits saving: reduce error to 1 poly(r) with only r original random bits. Expander codes: construction for good codes. Metric embedding: embed any N-point metric space into l2 metric with only log(N) distortion and dimension at most log(N) guranteed, expanders happen to be some of the worst case example. Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 4 / 8
. Theorem 2. . . . . . . . . Fix d ≥ 3, a random d-regular graph is a (Ω(N), d − 1.01)-expander WHP. Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 5 / 8
∀α. A random d-regular graph has spectral expansion at most 2 √ d WHP, which means vertex expansion at least d 4 . Vertex => Spectral. Loosely equivalent to vertex expansion But not in the optimal case: Ramanujan graphs. Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 6 / 8
. . . . . . . G is connected, non-bipartite, multi-digraph on n vertices, then λ < 1 and mixing time is O(log n 1−λ ). . Lemma 4. . . . . . . . . 1 − λ ≥ 1 dn2 and thus G has mixing time O(dn2 log n). Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 7 / 8