Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Expanders and L vs. RL

Expanders and L vs. RL

Algorithm class presentation

Jingcheng Liu

December 10, 2012
Tweet

More Decks by Jingcheng Liu

Other Decks in Education

Transcript

  1. . . . . . . . Expanders and L

    vs. RL Jingcheng Liu, Chengyu Lin ACM 2010 December 10, 2012 Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 1 / 8
  2. Outline . . . 1 Vertex Expansion . . .

    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
  3. Vertex Expansion Sparse but yet well connected. . Definition 1.

    . . . . . . . . (K,A)-expander: ∀S ⊆ V, ∥S∥ ≤ K, N(S) ≥ A · ∥S∥ Desirable: D = O(1), K = Ω(N), A = Ω(D). Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 3 / 8
  4. Vertex Expansion Hardness of linear transformation: counter-example for superconcentrator with

    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
  5. Existence of good expanders Just pick a random d-regular graph.

    Jingcheng Liu, Chengyu Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 5 / 8
  6. Existence of good expanders Just pick a random d-regular graph.

    . 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
  7. Spectral Expansion Spectral λ=> Vertex expansion (αN, 1 (1−α)λ2+α ),

    ∀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
  8. Bounds on random walks Mixing time. . Lemma 3. .

    . . . . . . . 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
  9. L vs. RL and Zig zag expanders Jingcheng Liu, Chengyu

    Lin (ACM 2010) Expanders and L vs. RL December 10, 2012 8 / 8