Transcript The zigzag product, Expander graphs & Combinatorics vs

The zigzag product,
Expander graphs &
Combinatorics vs. Algebra
Avi Wigderson
IAS & Hebrew University
’00 Reingold, Vadhan, W.
’01 Alon, Lubotzky, W.
’01 Capalbo, Reingold, Vadhan, W.
’02 Meshulam, 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 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!
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 - …
• …
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 = SL2(p) : group 2 x 2 matrices of det 1 over Zp.
S = { M1 , M2 } : M1 = ( 10 11 ) , M2 = ( 11 01 )
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]
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)
Explicit Constructions (Combinatorial)
-Zigzag Product [RVW]
G an [n, m, ]-graph. H an [m, d, ]-graph.
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.
H
Example
G=B2m, the Boolean m-dim cube ([2m,m]-graph).
H=Cm , the m-cycle ([m,2]-graph).
G z H is the cube-connected-cycle ([m2m,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,d2,+]-graph.
Proof: Follows simple information theoretic intuition.
The construction:
Start with a constant size H a [d4,d,1/4]-graph.
• G1 = H 2
• Gk+1 = Gk2 z’ H
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.
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: 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
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
+
0
+
0
0
+
1
0
+
0
Pz
+
0
C is a linear code
Rate (C)  k/n , Encoding time = O(n2)
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
1
+
1
Pw
+
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 |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 = sB (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=F2m, the vector space, S={e1, e2, …, em} , the unit vectors
B=Zm, the cyclic group, T={1}, shift by 1
B acts on A by shifting coordinates. S=e1B.
G =C(A,S), H = C(B,T), and
G z H = C(A x B, {e1 }  {1 } )
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…)
Dimensions of Representations in
Expanding Groups [MW]
G naturally acts on FqG
(|G|,q)=1
Assume: G is expanding
Want: G x FqG expanding
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.
Lemma. If G is monomial, so is G x FqG
Iterative construction of near-constant degree
expanding Cayley graphs
Iterate: G’ = G x FqG
Start with G1 = Z3
Get G1 , G2,…, Gn ,…
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!
Open Questions
Explicit undirected, const degree, lossless expanders
 Expanding Cayley graphs of constant degree “from scratch”.
 Construct expanding generators with few orbits
(= highly symmetric linear codes)
- In other group actions.
- Explicit instead of probabilistic.
 Prove or disprove: every expanding group G has < exp (d)
irreducuble representations of dimension d.
 Are SL2(p) always expanding?
 Are Sn never expanding?
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