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

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!

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

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

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

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

Algebraic explicit constructions [M,GG,AM,LPS,L,…] Many such constructions are Cayley graphs. 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 SL 2 (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] A = SL 2 (p) : group 2 x 2 matrices of det 1 over Z p. S = { M 1, M 2 } : M 1 = ( ), M 2 = ( )

Is expantion a group property? A constant number of generators. Annoying questions:  non-expanding generators for SL 2 (p)?  Expanding generators for the family S n ?  expanding generators for Z n ? No! [K] Basic question [LW]: Is expansion a group property? Is C(G i,S i ) an expander family if C(G i,S i ’) is? Theorem. [ALW] No!! Note: Easy for nonconstant number of generators: C(F 2 m,{e 1, e 2, …,e m }) is not an expander (This is just the Boolean cube) But  v 1,v 2, …,v 2m for which C(F 2 m,{v 1,v 2, …,v 2m }) is an expander (This is just a good linear error-correcting code)

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

Example G=B 2 m, the Boolean m-dim cube ([2 m,m]-graph). H=C m, the m-cycle ([m,2]-graph). m=3 G z H is the cube-connected-cycle ([m2 m,3]-graph)

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 [d 4,d,1/4]-graph. G 1 = H 2 Theorem. [RVW] G k is a [d 4k, d 2, ½]-graph. Proof: G k 2 is a [d 4k,d 4, ¼]-graph. H is a [d 4, d, ¼]-graph. G k+1 is a [d 4(k+1), d 2, ½]-graph. A stronger product z’ : Theorem. [RVW] G z’ H is an [nm,d 2,  +  ]-graph. G k+1 = G k 2 z’ H

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. Minimize d Ramanujan graphs: d=  (a 2 ) Random graphs: d=O(a log a) Zig-zag graphs: [RVW] d=O(a(log a) O(1) ) Uses zig-zag product on extractors!

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

Error Correcting Codes [Shannon, Hamming] C: {0,1} k  {0,1} n C=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,…] z  C iff Pz=0 C is a linear code Trivial Rate (C)  k/n, Encoding time = O(n 2 ) G lossless  Dist (C) =  (n), Decoding time = O(n) n n-k z Pz

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

Semi-direct Product of groups A, B groups. B acts on A as automorphisms. Let a b denote the action of b on a. Definition. A  B has elements {(a,b) : a  A, b  B}. group mult (a’,b’) (a,b) = (a’a b, b’b) Main Connection Assume = B, = A, S = s B (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) = (as b,b) (Step between clouds) Extends to more orbits Theorem [ALW] C(A x B, {s}  T ) = C (A,S ) z C (B,T )

Example A=F 2 m, the vector space, S={e1, e2, …, em}, the unit vectors B=Z m, the cyclic group, T={1}, shift by 1 B acts on A by shifting coordinates. S=e 1 B. G =C(A,S), H = C(B,T), and Expansion is not a group property! [ALW] C(A, e 1 B ) is not an expander. C(A, u B  v B ) is an expander for most u,v  A. [MW] C(A x B, {e1 }  {1 } ) is not an expander. C(A x B, {u B  v B }  {1 } ) is an expander (almost…) G z H = C(A x B, {e 1 }  {1 } )

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

Iterative probabilistic construction of near-constant degree expanding Cayley graphs Iterate: G’ = G x F q G Start with G 1 = Z 3 Get G 1, G 2,…, G n,… Exist S 1, S 2,…, S n,… = G n Theorem. [MW] (C(G n, S n ))  ½ (expanding Cayley graphs) |S n |  O(log (n/2) |G n |) (deg “approaching” constant) Theorem [LW] This is tight!

Iterative explicit construction of constant degree expanding Cayley graphs (under assumption) Iterate: G’ = G wr A k Start with G 1 = A k Get G 1, G 2,…, G n,… Construct S 1, S 2,…, S n,… Theorem. [RSW] (C(G n, S n ))  ½ (expanding Cayley graphs) |S n |  k (explicit, constant degree) Assumption A k can be made expanding with k  generators

Open Questions  Expanding Cayley graphs of constant degree “from scratch”.  Prove or disprove: every expanding group G has < exp (d) I irreducible representations of dimension d.  Are SL 2 (p) always expanding with every constant gen set?  Are S k expanding with some constant generating set? with some size k  generating set?  Explicit undirected, const degree, lossless expanders