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A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product

A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product. Avraham Ben-Aroya Amnon Ta-Shma Tel-Aviv University. Expander graphs. Sparse graphs with strong connectivity. Fundamental objects in Math and CS Communication networks Derandomization

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A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product

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  1. A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Amnon Ta-Shma Tel-Aviv University

  2. Expander graphs Sparse graphs with strong connectivity. Fundamental objects in Math and CS • Communication networks • Derandomization • Error correcting codes • Proof complexity • … Challenge: Explicit constructions of “good” expanders

  3. Spectral gap 1 = 1 2 3 n spectrum D-regular graph operator • Eigenvector basis v1,…,vn, eigenvalues 1=1  …n • Denote 2(G)=max{|2|,|n|}. • Smaller 2(G)  better expansion

  4. Bounds • Alon Boppana: Any D-regular graph satisfies 2(G)  2(D-1) /D – o(1)  2D-1/2 o(1)0 as the number of vertices grows • Friedman: Random D-regular graph are (w.h.p.) almost Ramanujan 2(G)  2(D-1) /D .

  5. Expander constructions

  6. Why do we look for combinatorial constructions? • A fundamental object in CS deserves a combinatorial proof [RVW00] • Central component in: • Good combinatorial expanders [CRVW02] • Undirected connectivity in L [Reingold05] • Cayley expanders [ALW01, MW02, RSW04]

  7. Expander constructions Full= The neighbors of a vertex are computable in time polylog(|V|)

  8. Expander constructions Mild= The graph is computable in time poly(|V|)

  9. Expander constructions

  10. Outline • The result • The technique: a new variant of the zig-zag product

  11. Heart of [ReingoldVadhanWigderson00] the zig-zag product G zig-zag (N,DG,G) G zigzag H H (DG,DH,H) (N·DG, DH2, ≈H+G) 2 degree vertices

  12. The replacement product

  13. The replacement product Cloud

  14. The Zig-Zag product • Vertices: same as in replacement product • Edges: (v,u)E there is a path of length 3 on the replacement product such that: • The first step is inside a cloud • The second step is inter-cloud • The third step is inside a cloud

  15. The Zig-Zag product Example: v and u are connected u v

  16. Why does the zig-zag works? The graph after replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud

  17. Why does the zig-zag works? Wasted H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud A uniform distribution over a subset of clouds

  18. Why does the zig-zag works? H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  19. Why does the zig-zag works? H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  20. Spectral gap of the zig-zag • We take two H-steps, hence the degree is DH2. • On some inputs, only one of the two steps is useful  This incurs a quadratic loss in 2.

  21. u v First Attempt: zig-zag with 3 cloud steps

  22. The problem The graph after replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud

  23. The problem Wasted H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud A uniform distribution over a subset of clouds

  24. The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  25. cloud The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud

  26. Ideally: H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  27. cloud The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud

  28. The problem H step G step H step G step H step cloud cloud cloud Wasted cloud cloud cloud cloud cloud cloud Potentially: 1 out of 2 cloud steps is wasted

  29. Our solution • Once an H-step is wasted, all the following H-steps are not. • We shall make sure that all the following G-steps are cloud-dispersing. • This is done by taking thicker clouds and choosing H in a special way.

  30. Replacement with thicker clouds An expander over the cloud vertices

  31. Replacement with thicker clouds Arbirary matching

  32. Replacement with thicker clouds Arbirary matching

  33. Why does this work? The graph after modified replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud D10 vertices

  34. Why does this work? Wasted H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud A uniform distribution over a subset of clouds D10 vertices

  35. Why does this work? D9 vertices H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud D10 vertices

  36. Why does this work? Almost uniform over outgoing edges H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  37. Why does this work? H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

  38. Why does this work? H step G step H step G step H step cloud cloud cloud Not Wasted cloud cloud cloud cloud cloud cloud

  39. A key point We show that for random graphs Hi after applying • A cloud step • An inter cloud step • A cloud step we are almost uniform over the D outgoing edges.

  40. A problem The graphs Hi are of constant size, The number of clouds = The size of the large graph and grows to infinity. It seems unlikely that there is a choice for Hi that is good for all initial distributions.

  41. A solution Take the large graph G to be “locally invertible” G(v,i)=(v[i],(i)) Then, the labels just depend on the V2 component, :V2 -> V2, Hi A standard probabilistic method argument works.

  42. Conclusions • Our result: a fully-explicit combinatorial construction of D-regular graphs with 2(G)=D-1/2+o(1) . • Can we push this further to O(D-1/2) ? Ramanujan+ ?

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