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 Applications in: • Communication networks • Derandomization • Error correcting codes • PCPs • Proof complexity • …

3. Properties of expanders Many pseudorandom properties: • No small cuts • Every “not-too-large” set expands • Random walks mix fast • Spectral gap • … Challenge: Explicit constructions of “good” expanders

4. 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

5. What is the optimal spectral gap? • [AlonBoppana] : 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 graphs satisfy 2(G)  2(D-1) /D +  • A graph is Ramanujan if 2(G)  2(D-1) /D

6. Expander constructions

7. 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]

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

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

10. Expander constructions

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

12. The construction scheme [ReingoldVadhanWigderson00] Small expander Increase size Reduce degree Improve 2

13. 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

14. The replacement product

15. The replacement product Slightly incomplete….

16. The replacement product For simplicity assume G is DG edge-colorable

17. The replacement product Cloud

18. 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

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

20. Intuition vs. formality • Formally: spectral analysis – not in this talk • Intuitively: analyze “entropy flow” Entropy- a measure of randomness (H2()=-log Prx,y~[x=y]) If G is D-regular then H2(G())H2()+log(D) When G has a good spectral gap this almost tight

21. Intuition vs. formality • Intuitive analysis: start with some - a entropy deficient distribution on its vertices • Show that H2(G())>>H2() • We’ll analyze only one illuminating distribution • Formal analysis follows a similar argument

22. Why does the zig-zag work? The graph after replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud

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

24. Why does the zig-zag work? H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

25. Why does the zig-zag work? H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

26. 65 65 65 7 10 3 Why does the zig-zag work?Buffer point-of-view A vertex in the new graph: Cloud-label Edge-label [N][DG] A H-step: 0.5 0.5 H 10 3 7

27. 65 9 3 3 Why does the zig-zag work?Buffer point-of-view A vertex in the new graph: Cloud-label Edge-label [N][DG] A G-step: G For simplicity assume G is DG edge-colorable 3 65 9

28. A uniform distribution over a subset of clouds: • The marginal on is uniform • Conditioned on every cloud in the support, is uniform Why does the zig-zag work?Buffer point-of-view H step G step H step The support of the distribution Cloud-label Edge-label [N][DG]

29. Why does the zig-zag work?Buffer point-of-view Wasted H step G step H step Cloud-label Edge-label Cloud=k Uniform H does nothing

30. Why does the zig-zag work?Buffer point-of-view H step G step H step Cloud-label Edge-label Edge-label - Applying G is just taking a step fromaccording to - Since G is an expander, after this step, should have more entropy - G is a permutation, hence the entropy in is reduced

31. Why does the zig-zag work?Buffer point-of-view H step G step H step Cloud-label Edge-label Cloud=k H adds entropy!

32. 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.

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

34. The problem The graph after replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud

35. 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

36. The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

37. cloud The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud

38. Ideally: H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud cloud

39. cloud The problem H step G step H step G step H step cloud cloud cloud cloud cloud cloud cloud cloud

40. 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

41. Cloud=k Uniform H does nothing The problem:Buffer point-of-view H step G step H step G step H step Cloud-label Edge-label A uniform distribution over a subset of clouds

42. The problem:Buffer point-of-view H step G step H step G step H step Cloud-label Edge-label Edge-label - Applying G is just taking a step fromaccording to - Since G is an expander, after this step, has more entropy - G is a permutation, hence the entropy in is reduced

43. The problem:Buffer point-of-view H step G step H step G step H step Cloud-label Edge-label Cloud=k H adds entropy!

44. The problem:Buffer point-of-view H step G step H step G step H step Cloud-label Edge-label Edge-label ? ? Since the second component is not uniform, we don’t know how the entropy flows in the G step (this is like taking a random step over the graph according to an unknown distribution).

45. The problem:Buffer point-of-view H step G step H step G step H step Cloud-label Edge-label Cloud=k Uniform We might be in this case, in which H does nothing Potentially: 1 out of 2 cloud steps is wasted

46. 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

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

48. Replacement with thicker clouds Multiple parallel edges

49. Why does this work? The graph after modified replacement cloud cloud cloud cloud cloud cloud cloud cloud cloud DG10 vertices

50. 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