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

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+ ?