The zigzag product, Expander graphs

1
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.
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 v1 , v ? 1

?(G) ? 0,1
Definition. Gi is an expander family if ?(Gi)?
?lt1
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)? ?.
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
Deterministic amplification
Prerror lt 1/3
Thm Chernoff r1 r2. rk independent (kn
random bits)
Thm AKS r1 r2. rk random path (n O(k)
random bits)
then Prerror Prr1 r2. rk ?Bx
gt k/2 lt exp(-k)
7
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 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, ,v2m for which C(F2m,v1,v2, ,v2m)
is an expander
(This is just a good linear error-correcting code)
9
Explicit Constructions (Combinatorial)-Zigzag
Product RVW
G an n, m, ?-graph. H an m, d, ?-graph.
Combinatorial construction of expanders.
10
Example
GB2m, the Boolean m-dim cube (2m,m-graph).
HCm , the m-cycle (m,2-graph).
m3
11
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.
Gk1 is a d 4(k1), 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.
Minimize d
Ramanujan graphs d?(a2)
Random graphs dO(a log a)
Zig-zag graphs RVW dO(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
Error Correcting Codes Shannon, Hamming
C 0,1k ? 0,1n CIm(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
0 Pz

1 1 0 1 0
0 1 1 z
z?C iff Pz0 C is a linear
code
Trivial Rate (C) ? k/n , Encoding time
O(n2)
G lossless ? Dist (C) ?(n), Decoding time
O(n)
15
Decoding
Thm CRVW Can explicitly construct
graphs kn/2, bottom deg 10, ?B?n, B?
n/200, ?(B) ? 9B
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 1s than 0s
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
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) (aab , bb)
Main Connection Assume ltTgt B, ltSgt 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
Example
AF2m, the vector space, Se1, e2, , em , the
unit vectors
BZm, the cyclic group, T1, shift by 1
B acts on A by shifting coordinates. Se1B.
G C(A,S), H C(B,T), and
Expansion is not a group property! ALW
C(A, e1B ) is not an expander.
C(A x B, e1 ? 1 ) is not an
expander.
C(A, u B?vB) is an expander for most u,v ?A.
MW
C(A x B, uB ?vB ? 1 ) is an expander
(almost)
18
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 ,
ltSn gt 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 G has
lt exp (d) I irreducible
representations of dimension d.