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An Elementary Construction of Constant-Degree Expanders

An Elementary Construction of Constant-Degree Expanders. Slides: Ofer Rothschild, 2009 Noga Alon Oded Schwartz Asaf Shapira Tel Aviv University, 2007. Expanders. Every cut contains (relatively) many edges A d-regular graph is a δ -edge-expander,

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An Elementary Construction of Constant-Degree Expanders

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  1. An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009 Noga Alon Oded Schwartz Asaf Shapira Tel Aviv University, 2007 An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  2. Expanders • Every cut contains (relatively) many edges • A d-regular graph is a δ-edge-expander, • if for every set S ⊆ V of size at most |V|/2 there are at least δd|S| edges connecting S and S = V \ S, • that is, e(S, V \ S) >= δd|S|. • [n,d, δ]-expander • (Why size at most |V|/2?) An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  3. Previous Results • Everywhere on the paper • Here they are denoted “PR” An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  4. Main Result • A polynomial time constructible constant-degree expander An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  5. Notes • Note that the largest eigenvalue of the adjacency matrix of any graph is d, with eigenvector 1n • The second eigenvalue will be denoted as λ2 • Next: a correlation between δ and λ2 An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  6. Theorem 1.1 PR • Let G be a δ-expander with adjacency matrix A and let λ2 =λ2(G) be the second-largest eigenvalue of A. • Then, 1/2 (1 − λ2/d) <= δ <=sqrt[2(1 − λ2/d)]. An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  7. Theorem 1.2 PR • There exists a fixed δ > 0, such that, for any d>=3 and even integer n, there is an [n, d, δ]-expander which is d-edge-colourable.1 • 1 d-edge-colourable: d colours, vertices don’t have 2 edges with the same colour • δ is known An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  8. Definition 1 – The Replacement Product (~120 words  ) • Let G be a D-regular D-edge-colourablegraph on n vertices and let H be a d-regular graph on D vertices. • Suppose G is already equipped with a proper D-edge-colouring. The replacement product G ◦ H is the following 2d-regular graph on nD vertices: • We first replace every vertex vi of G with a cluster of D vertices, which we denote Ci = {vi1, . . . , viD }. • For every 1<=i<=n we put a copy of H on Ciby connecting vipto viqif and only if (p, q) ∈ E(H). • Finally, for every (p, q) ∈ E(G) which is coloured t, we put d parallel edges between vpt and vqt. An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  9. For Example (n, D) graph G (D,d) graph H Example from: http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  10. Example from: http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  11. Example from: http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  12. Example from: http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  13. Example from: http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  14. Theorem 1.3 • Suppose E1 is an [n,D, δ1]-expander and E2 is a [D, d, δ2]-expander. Then, E1 ◦ E2 is an [nD, 2d, (1/80)δ12δ2]-expander. • Proof sketch: Let X ⊆ V of size at most ½nD • Most Ci are sparsely populated  E2 expansion • Most Ci are densely populated  E1 expansion • Algebraic mumbo-jumbo An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  15. Theorem 1.4 – Main Result • There exists a fixed δ > 0 such that for any integer q = 2t and for any q4/100<=r<=q4/2 there is a polynomial-time constructible [q4r+12, 12, δ]-expander • Proof sketch: • Building some expanders and multiplying them An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  16. 2. The Construction • q=2t , r natural. LD(q,r): • Vertices: The elements of Fqr+1 : • (a0, . . . , ar) ∈ Fqr+1 , ai∈ Fq: • (ai1,…,ait) ∈ Fq , aij∈ Z2, • Addition: coordinate-by-coordinate • Product: not important • For every vertex, a neighbor for every (x,y) in Fq2 : • a + y · (1, x, x2, . . . , xr ) • LD(q,r) is q2-regular and q2-edge-colourable An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  17. Example: LD(2,2) • q=21 , r=2, Fr+1 =Z2 3 (1,0,0) (1,1,0) (0,0,0) (0,1,0) (0,0,1) (0,1,1) (1,0,1) (1,1,1) Neighbors: For all (x,y): a+y(1,x,x2 ) An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  18. Theorem 2.1 • For any q = 2t and integer r < q we have λ2(LD(q, r))<=rq. • Proof: • We will find the eigenvectors of the adjacency matrix M and show λ2<=rq An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  19. Proof of Theorem 2.1 (cont.) • For every a ∈ Fr+1, let va be the vector whose bth entry (where b ∈ Fr+1) satisfies va(b) = (−1)L(<a,b>). • {va} are orthogonal • va(b + c) = va(b)va(c) An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  20. Proof of Theorem 2.1 (cont.) • If pa(x) = 0, then (−1)L(y·pa(x)) = 1 for all y, thus such an x contributes q to λa • If pa(x) ≠ 0 then y · pa(x) takes on all values in F as y varies, and hence (−1)L(y·pa(x)) varies uniformly over {−1, 1}, implying that these xs contribute nothing to λa An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  21. Proof of Theorem 2.1 (cont.) • Therefore, when a = 0n we have λa = λ1 = q2 • Otherwise, when a ≠ 0n, pa has at most r roots, and therefore λa<=rq • Therefore, λ2<=rq • Quod erat demonstrandum An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  22. Concluding remarks • Expanders on Θ(n) vertices • Fully explicit expanders • Edge expansion close to ½ • Eigenvalue gap • Expanders with smaller degree An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

  23. References • NogaAlon, Oded Schwartz, and AsafShapira.An elementary construction of constant-degree expanders.Combin. Probab. Comput., 17(3):319-327, 2008. http://www.cs.tau.ac.il/courses/combsem/09a/papers/Expanders.pdf • Manor Mendel and AssafNaor. Towards a calculus for non-linear spectral gaps. 2008. http://sites.google.com/site/mendelma/Home/research/zigzag-tau-comb.pdf An Elementary Construction of Constant-Degree Expanders Slides: Ofer Rothschild, 2009

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