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The zigzag product, Expander graphs & Combinatorics vs. Algebra

Avi Wigderson IAS, Princeton. The zigzag product, Expander graphs & Combinatorics vs. Algebra. ’00 Reingold, Vadhan, W. ’01 Alon, Lubotzky, W. ’01 Capalbo, Reingold, Vadhan, W. ’02 Meshulam, W. ’03 Rozenman, Shalev, W. Expanding Graphs - Properties.

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The zigzag product, Expander graphs & Combinatorics vs. Algebra

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  1. Avi Wigderson IAS, Princeton The zigzag product,Expander graphs & Combinatorics vs. Algebra ’00 Reingold, Vadhan, W. ’01 Alon, Lubotzky, W. ’01 Capalbo, Reingold, Vadhan, W. ’02 Meshulam, W. ’03 Rozenman, Shalev, W.

  2. Expanding Graphs - Properties • Combinatorial: no small cuts, high connectivity • Probabilistic: rapid convergence of random walk • Algebraic: small second eigenvalue Theorem. [C,T,AM,A,JS] All properties are equivalent!

  3. Expanders - Definition Undirected, regular (multi)graphs. Definition. The 2nd eigenvalue of a d-regular G (G) = max { || (AG /d) v || : ||v||=1 , v  1 } (G)  [0,1] Definition. {Gi}is an expander family if (Gi) <1 Theorem [P]Most 3-regular graphs are expanders. Challenge: Explicit (small degree) expanders! Gis [n,d]-graph: nvertices, d-regular Gis [n,d, ]-graph: (G) .

  4. Applications of Expanders In CS • Derandomization • Circuit Complexity • Error Correcting Codes • Communication Networks • Approximate Counting • Computational Information • …

  5. Applications of Expanders In Pure Math • Topology – expanding manifolds [Br,G] • Group Theory – generating random gp elements [Ba,LP] • Measure Theory – Ruziewicz Problem [D,LPS], • F-spaces [KR] • Number Theory – Thin Sets [AIKPS] • Graph Theory - … • …

  6. Bx G [2n,d,1/8]-graph G explicit! {0,1}n random strings r rk r1 x x x Alg Alg Alg Majority Deterministic amplification Pr[error] < 1/3 Thm [Chernoff] r1 r2….rkindependent (kn random bits) Thm [AKS] r1 r2….rkrandom path (n+ O(k) random bits) then Pr[error] = Pr[|{r1 r2….rk }Bx}| > k/2] < exp(-k)

  7. A = SL2(p): group 2 x 2 matrices of det 1 over Zp. S = { M1 , M2 } : M1 = ( ) , M2 = ( ) 1 1 0 1 1 0 1 1 Algebraic explicit constructions [M,GG,AM,LPS,L,…] Many such constructions are Cayleygraphs. A a finite group, S a set of generators. Def. C(A,S) has vertices A and edges (a, as) for all aA, sSS-1. Theorem. [L]C(A,S) is an expander family. Proof: “The mother group approach”: • Use SL2(Z)to define a manifold N. • Bound the e-value of (the Laplacian of) N [Sel] • Show that the above graphs “well approximate” N. Works with any finite generating set, other groups, group actions… Theorem. [LPS,M] Optimal d(G) = 2(d-1) [AB]

  8. Is expantion a group property? A constant number of generators. Annoying questions: •  non-expanding generators for SL2(p)? •  Expanding generators for the family Sn? •  expanding generators for Z n? No! [K] Basic question [LW]:Is expansion a group property? Is C(Gi,Si) an expander family if C(Gi,Si’) is? Theorem. [ALW]No!! Note: Easy for nonconstant number of generators: C(F2m,{e1, e2, …,em}) is not an expander (This is just the Boolean cube) But v1,v2, …,v2mfor which C(F2m,{v1,v2, …,v2m})is an expander (This is just a good linear error-correcting code)

  9. H Definition. G zHhas vertices {(v,k) : vG, kH}. (v,k) v u Edges in clouds between clouds u-cloud v-cloud Theorem. [RVW]G zHis an [nm,d+1,f(,)]-graph, and <1, <1  f(,)<1. G zH is an expander iff Gand Hare. Explicit Constructions (Combinatorial)-Zigzag Product [RVW] Gan [n, m, ]-graph. H an [m, d, ]-graph. Combinatorial construction of expanders.

  10. G zHis the cube-connected-cycle ([m2m,3]-graph) Example G=B2m, the Boolean m-dim cube ([2m,m]-graph). H=Cm , the m-cycle ([m,2]-graph). m=3

  11. A stronger product z’: Theorem. [RVW]G z’ H is an [nm,d2,+]-graph. • Gk+1 = Gk2 z’ H Iterative Construction of Expanders G an [n,m,]-graph. H an [m,d,] -graph. Proof: Follows simple information theoretic intuition. The construction: Start with a constant size H a [d4,d,1/4]-graph. • G1 = H 2 Theorem. [RVW]Gk is a [d4k, d2, ½]-graph. Proof: Gk2 is a [d 4k,d 4, ¼]-graph. H is a [d 4, d, ¼]-graph. Gk+1 is a [d 4(k+1), d 2, ½]-graph.

  12. Beating e-value expansion In the following a is a large constant. Task:Construct an [n,d]-graph s.t. every two sets of size n/a are connected by an edge. Minimized Ramanujan graphs: d=(a2) Random graphs: d=O(a log a) Zig-zag graphs: [RVW]d=O(a(log a)O(1)) Uses zig-zag product on extractors!

  13. Lossless expanders [CRVW] Task: Construct an [n,d]-graph in which every set of size at most n/a expands by a factor c. Maximize c. Upper bound: cd Ramanujan graphs: [K]c  d/2 Random graphs: c  (1-)d Lossless Zig-zag graphs: [CRVW]c  (1-)d Lossless Use zig-zag product on conductors!! Extends to unbalanced bipartite graphs. Applications (where the factor of 2 matters): Data structures, Network routing, Error-correcting codes

  14. n-k n Error Correcting Codes [Shannon, Hamming] C: {0,1}k {0,1}nC=Im(C) Rate (C) = k/n Dist (C) = min d(C(x),C(y)) C good if Rate (C) = (1), Dist (C) = (n) Find good, explicit, efficient codes. Graph-based codes [G,M,T,SS,S,LMSS,…] 0 0 0 0 0 0Pz + + + + + + 1 1 0 1 0 0 1 1 z zC iff Pz=0 C is a linear code Trivial Rate (C)  k/n ,Encoding time = O(n2) Glossless  Dist (C) = (n), Decoding time = O(n)

  15. n-k n Decoding Thm [CRVW]Can explicitly construct graphs: k=n/2, bottom deg = 10, B[n], |B| n/200, |(B)|  9|B| 0 0 1 0 1 1 Pw + + + + + + 1 1 1 0 1 0 1 1 w Decoding alg [SS]: while Pw0 flip all wi with i in FLIP = { i : (i) has more 1’s than 0’s } B = set of corrupted positions |B|  n/200 B’ = set of corrupted positions after flip Claim [SS] : |B’|  |B|/2 Proof: |B \ FLIP |  |B|/4, |FLIP \ B |  |B|/4

  16. Theorem [ALW]C(A x B, {s}T ) = C (A,S ) z C (B,T ) Semi-direct Product of groups A,Bgroups. B acts on A as automorphisms. Let ab denote the action of bon a. Definition. A B has elements {(a,b) : aA, bB}. group mult(a’,b’) (a,b)= (a’ab, b’b) Main Connection Assume <T> = B, <S> = A , S = sB(S is a single B-orbit) Large expanding Cayley graphs from small ones. Proof: (of Thm) (a,b)(1,t) = (a,bt) (Step in a cloud) (a,b)(s,1) = (asb,b) (Step between clouds) Extends to more orbits

  17. G zH = C(A xB, {e1 } {1 } ) Example A=F2m, the vector space, S={e1, e2, …, em} , the unit vectors B=Zm, the cyclic group, T={1}, shift by 1 Bacts on A by shifting coordinates. S=e1B. G =C(A,S), H = C(B,T),and Expansion is not a group property! [ALW] C(A, e1B ) is notan expander. C(A xB, {e1 } {1 } )is not an expander. C(A, u BvB) is an expander for most u,v A. [MW] C(A xB, {uBvB } {1 } )is an expander (almost…)

  18. FqG expands with constant many orbits Thm 1 G has at most exp(d) irreducible reps of dimension d. Thm 2 G is expanding and monomial. Dimensions of Representations in Expanding Groups [MW] G naturally acts on FqG (|G|,q)=1 Assume: G is expanding Want: G x FqG expanding Lemma. If G is monomial, so is G x FqG

  19. Iterative probabilistic construction of near-constant degree expanding Cayley graphs Iterate: G’ = G x FqG Start with G1 = Z3 Get G1 , G2,…, Gn ,… Exist S1 , S2,…, Sn ,… <Sn > = Gn Theorem. [MW] (C(Gn, Sn))  ½ (expanding Cayley graphs) |Sn|  O(log(n/2)|Gn|) (deg “approaching” constant) Theorem [LW] This is tight!

  20. Iterative explicit construction of constant degree expanding Cayley graphs (under assumption) Iterate: G’ = G wr Ak Start with G1 = Ak Get G1 , G2,…, Gn ,… Construct S1 , S2,…, Sn ,… Theorem. [RSW] (C(Gn, Sn))  ½ (expanding Cayley graphs) |Sn|  k (explicit, constant degree) Assumption Ak can be made expanding with kgenerators

  21. Open Questions • Are Sk expanding with some constant generating set? with some size kgenerating set?  Are SL2(p) always expanding with every constant gen set? Expanding Cayley graphs of constant degree “from scratch”. Explicit undirected, const degree, lossless expanders Prove or disprove: every expanding group Ghas <exp (d) I irreducible representations of dimension d.

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