The zigzag product Expander graphs Combinatorics vs Algebra

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 (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! G is [n, d]-graph: n vertices, d-regular G is [n, d, ]-graph: (G) .

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

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 - … • …

Deterministic amplification G [2 n, d, 1/8]-graph G explicit! {0, 1} Bx random strings Pr[error] < 1/3 r 1 x Alg rk r x n Alg x Alg Majority Thm [Chernoff] r 1 r 2…. rk independent (kn random bits) Thm [AKS] r 1 r 2…. rk random path (n+ O(k) random bits) then Pr[error] = Pr[|{r 1 r 2…. rk } Bx}| > 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. A = SL 2(p) : group 2 x 2 matrices of det 1 over Zp. S = { M 1 , M 2 } : M 1 = ( 10 11 ) , M 2 = ( 11 01 ) 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]

Is expantion a group property? A constant number of generators. Annoying questions: • non-expanding generators for SL 2(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(F 2 m, {e 1, e 2, …, em}) is not an expander (This is just the Boolean cube) But v 1, v 2, …, v 2 m for which C(F 2 m, {v 1, v 2, …, v 2 m}) 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. H Definition. G z H has vertices {(v, k) : v G, k H}. v-cloud v (v, k) Edges u u-cloud in clouds between clouds Theorem. [RVW] G z H is an [nm, d+1, f( , )]-graph, and <1, <1 f( , )<1. G z H is an expander iff G and H are. Combinatorial construction of expanders.

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

Iterative Construction of Expanders A stronger product z’ : G an [n, m, ]-graph. H an [m, d, ] -graph. Theorem. [RVW] G z’ H is an [nm, d 2, + ]-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 • Gk+1 = Gk 2 z’ H Theorem. [RVW] Gk is a [d 4 k, d 2, ½]-graph. Proof: Gk 2 is a [d 4 k, d 4, ¼]-graph. H is a [d 4, d, ¼]-graph. Gk+1 is a [d 4(k+1), d 2, ½]-graph.

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 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 Rate (C) = k/n C=Im(C) 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 + n-k n 1 1 z C iff Pz=0 Trivial 0 0 + 1 0 + 0 Pz + 0 C is a linear code Rate (C) k/n , Encoding time = O(n 2) G lossless Dist (C) = (n), Decoding time = O(n) 1 1 z

Decoding Thm [CRVW] Can explicitly construct graphs: k=n/2, bottom deg = 10, B [n], |B| n/200, | (B)| 9|B| 0 + n-k n 1 1 0 1 + 0 + 1 1 + 0 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 Pw |B| n/200 B’ = set of corrupted positions after flip Claim [SS] : |B’| |B|/2 Proof: |B FLIP | |B|/4, |FLIP B | |B|/4 1 1 w

Semi-direct Product of groups A, B groups. B acts on A as automorphisms. Let ab 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’ab , b’b) Main Connection Assume <T> = B, <S> = A , S = s. B (S is a single B-orbit) Theorem [ALW] C(A x B, {s} T ) = C (A, S ) z C (B, T ) 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

Example A=F 2 m, the vector space, S={e 1, e 2, …, em} , the unit vectors B=Zm, 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 G z H = C(A x B, {e 1 } {1 } ) Expansion is not a group property! [ALW] C(A, e 1 B ) is not an expander. C(A x B, {e 1 } {1 } ) is not an expander. C(A, u B v. B) is an expander for most u, v A. [MW] C(A x B, {u. B v. B } {1 } ) is an expander (almost…)

Dimensions of Representations in Expanding Groups [MW] G naturally acts on Fq. G (|G|, q)=1 Assume: G is expanding Want: G x Fq. G expanding Fq. G 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. Lemma. If G is monomial, so is G x Fq. G

Iterative probabilistic construction of near-constant degree expanding Cayley graphs Iterate: G’ = G x Fq. G Start with G 1 = Z 3 Get G 1 , G 2, …, Gn , … Exist S 1 , S 2, …, 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!

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

Open Questions Are Sk expanding with some constant generating set? with some size k generating set? Are SL 2(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 G has < exp (d) irreducible representations of dimension d. I