Transcript The zigzag product, Expander graphs & Combinatorics vs

Expander graphs – applications and
combinatorial constructions
Avi Wigderson
IAS, Princeton
[Hoory, Linial, W. 2006]
“Expander graphs and applications”
Bulletin of the AMS.
Applications
in Math & CS
Applications of Expanders
In CS
• Derandomization
• Circuit Complexity
• Error Correcting Codes
• Communication & Sorting Networks
• Approximate Counting
• Computational Information
• Data Structures
•…
Applications of Expanders
In Pure Math
• Topology – expanding manifolds [Brooks]
- Baum-Connes Conjecture [Gromov]
• Group Theory – generating random group elements
[Babai,Lubotzky-Pak]
• Measure Theory – Ruziewicz Problem [Drinfeld,
Lubotzky-Phillips-Sarnak], F-spaces [Kalton-Rogers]
• Number Theory Thin Sets [Ajtai-Iwaniec-Komlos-PintzSzemeredi] -Sieve method [Bourgain-Gamburd-Sarnak]
- Distribution of integer points on spheres [Venkatesh]
• Graph Theory - …
Expander graphs:
Definition and
basic properties
Expanding Graphs - Properties
• Combinatorial: no small cuts, high connectivity
• Probabilistic: rapid convergence of random walk
• Algebraic: small second eigenvalue
Theorem. [Cheeger, Buser, Tanner, Alon-Milman, Alon,
Jerrum-Sinclair,…]: All properties are equivalent!
Expanding Graphs - Properties
G(V,E)
S
V vertices, E edges
|V|=n (  ∞ )
d
d-regular (d fixed)
S |S|< n/2
|E(S,Sc)| > α|S|d (what we expect in a random graph)
α constant
• Combinatorial: no small cuts, high connectivity
• Geometric: high isoperimetry
Expanding Graphs - Properties
G(V,E)
d-regular
v1, v2, v3,…, vt,…
vk+1 a random neighbor of vk
vt converges to the uniform distribution
in O(log n) steps (as fast as possible)
• Probabilistic: rapid convergence of random walk
Expanding Graphs - Properties
G(V,E)
V
V
1 = 1 ≥ 2 ≥ … ≥ n ≥ -1
(G) = maxi>1 |i| =
max { AG v : v =1, vu }
(G) ≤  < 1
1-(G)
AG(u,v) =
AG
0 (u,v)  E
1/d (u,v)  E
normalized adjacency matrix
(random walk matrix)
“spectral gap”
• Algebraic: small second eigenvalue
Expanders - Definition
Undirected, regular (multi)graphs.
G is [n,d]-graph: n vertices, d-regular.
Theorem: An [n,d]-graph G is connected iff (G) <1.
Indeed, if G is (non-bip) connected than (G) <1- 1/dn2
G is [n,d,  ]-graph: (G)  . G expander if  <1.
Definition: An infinite family {Gi} of [ni,d, ]-graphs is
an expander family if for all i  <1 .
Random walk convergence
(algebraic exp.  fast mixing)
G(V,E)
V
V
1 = 1 ≥ 2 ≥ … ≥ n eigenvalues
u = v1
v2 …
vn eigenvectors
AG(u,v) =
AG
0 (u,v)  E
1/d (u,v)  E
p: any probability distribution on V
p(t)= (AG)tp: distribution after t steps of a random walk
p(t) – u1  n  p(t) – u  = n  (AG)t (p–u)   n ((G))t
Converges in O(log n) steps. Corollary: Diam (G) < O(log n).
Expander mixing lemma [Alon-Chung]
(algebraic exp.  combinatorial exp.)
G(V,E)
V
V
1 = 1 ≥ 2 ≥ … ≥ n eigenvalues
u = v1
v2 …
S V 1S = i ivi
vn eigenvectors
AG(u,v) =
AG
0 (u,v)  E
1/d (u,v)  E
1 = |S|/n
Expected cut size:
like a random graph
T V 1T = i ivi 1 = |T|/n
AG  J/n
1SAG1T  = i iii = |S||T|/n + i>1 iiI
||E(S,T)|- d|S||T|/n |  d(G)i>1ii  d(G) |S||T|  d(G)n
Existence,
explicit construction
and basic parameters
Existence of sparse expanders
Theorem [Pinsker] Most 3-regular graphs are expanders.
Proof: The probabilistic method (an early application).
Take a random 3-regular graph G(V,E) with |V|=n.
Fix a set S V, |S| = s < n/2.
Pr[|E(S,Sc)| < .1s] < (s/n)3s
Pr[ G is not expanding ] < s ( ns ) (s/n)3s < 1/2
Challenge: Explicit (small degree) expanders!
Explicit construction
G(V,E)
V
V
{Gi(Vi,Ei)} infinite sequence of graphs
AG
AG(u,v) =
0 (u,v)  E
1/d (u,v)  E
Weakly explicit:
AGi can be constructed in poly (|Vi|) time
Strongly explicit:
AGi can be constructed in polylog (|Vi|) time:
A poly(n) algorithm, given i, u  Vi, lists all v  Vi s.t. (u,v)  Ei
How small can (G) be?
Amplification: Consider Gk
V
V
AG
G is [n,d,  ]-graph (with  <1) then
Gk is [n,dk, k ]-graph
So increasing d we can make (G) = 1/dc for some c>0
Theorem: [Alon-Boppana] An infinite family {Gi} of
[ni,d]-graphs must have (G) > (2 (d-1))/d  1/d
Proof: (of a weaker statement). Assume d < n/2.
n/d = Tr[AGAGt] = i i2 < 1+(n-1)(G)2

1/2d < (G)
Basic consequences: G [n,d,]-graph
Theorem. If S,T V, (u,v)  E a random edge, then
| Pr[u S and v T] - (S) (T) | < 
Cor 1: A random neighbor of a random vertex in S will
land in S with probability < (S)  
Cor 2: Every set of size > n contains an edge.
 Chromatic number (G) > 1/
 Graphs of large girth and chromatic number
Cor 3: Removing any fraction  <  of the edges leaves a
connected component of 1-O() of the vertices.
Fault-tolerant computation
Infection Processes: G [n,d,]-graph, <1/4
Cor 4: Every set S of size s < n/2 contains at most s/2
vertices with > 2s neighbors in S
Infection process 1: Adversary infects I0, |I0|  n/4.
I0=S0, S1, S2, …St,… are defined by:
v  St+1 iff a majority of its neighbors are in St.
Fact: St= for t > log n
[infection dies out]
Proof: |Si+1|≤|Si|/2
Infection process 2: Adversary picks I0, I1,… , |It| n/4.
I0=R0, R1, R2, …Rt,… are defined by Rt = St  It
Fact: |Rt| n/2 for all t
[infection never spreads]
Reliable circuits from unreliable components
[von Neumann]
Given, a circuit C for f of size s
Every gate fails with prob p < 1/10
Construct C’ for C’(x)=f(x) whp.
V
X1
1
V
0
1
V
V
0
0
V
Possible? With small s’?
V
1
f
X2
X3
Reliable circuits from unreliable components
[von Neumann]
Given, a circuit C for f of size s
Every gate fails with prob p < 1/10
I
Construct C’ for C’(x)=f(x) whp.
V
f
Possible? With small s’?
I
V
- Add Identity gates
I
V
I
I
V
V
V
X2
X3
I
X1
Reliable circuits from unreliable components
[von Neumann]
Given, a circuit C for f of size s
I
I
I
V
V
V
V
I
I
I
I
I
I
V
V
V
V
V
V
V
V
V
V
V
I
V
I
V
I
V
X1 X1 X1 X1 X2 X2 X2 X2
V
I
V
I
I
V
V
I
V
V
I
I
I
-Reduce errors
I
I
-Replicate circuit
1
I
Possible? With small s’?
-Add Identity gates
f
I
Construct C’ for C’(x)=f(x) whp.
I
Every gate fails with prob p < 1/10
X3 X3 X3 X3
Reliable circuits from unreliable components
[von Neumann, Dobrushin-Ortyukov, Pippenger]
Given, a circuit C for f of size s
M
M
M
V
V
V
V
M
M
M
M
M
M
V
V
V
V
M
M
M
M
M
V
M
V
V
V
V
V
V
V
X1 X1 X1 X1 X2 X2 X2 X2
M
V
V
Process 2
M
V
Infection
M
V
Analysis:
V
of size O(log s) 
M
M
V
M
1
M
M
V
Possible? With small s’?
Majority “expanders”
f
M
Construct C’ for C’(x)=f(x) whp.
V
Every gate fails with prob p < 1/10
X3 X3 X3 X3
Bipartite expanders
(unbalanced) Bipartite expanders
G(V,E)
w
u
SV’ |S| n/2,
|NV’’ (S)| (3/2)|S|
V’
V’’
V’’’
H(V’,V’’’;E*)
|V’|=n, |V’’’|=(2/3)n, degree=4d
SV’ |S| n/2,
|NV’’’ (S)| |S|
 S has a perfect matching to V’’’
u’
w’
u”
w”
Concentrators Cn [Bassalygo,Pinsker]
Cn
V’
n
V’’
2n/3
SV’ |S| n/2,
S has a perfect matching to V’’
Networks
- Fault-tolerance
- Routing
- Distributed computing
- Sorting
Superconcentrators [Valiant]
|I|=|O|=n
 k n
SCn
 I’  I
 O’  O
|I’|=|O’|=k
There are k vertex
disjoint paths I’ to O’
I
How many edges are
needed for SCn ?
[V] This number is a
Circuit lower bound for
Disc. Fourier Transform
O
Superconcentrators & circuits [Valiant]
[V] |E(SCn)| is a
circuit lower bound for
linear circuits computing
any matrix A with all
minors nonsingular
X1 X2 X3
Xn
I


y = Ax

 k n I’  I O’  O
|I’|=|O’|=k there are k
Vertex disjoint paths I’ to O’
Otherwise, by Menger’s
Theorem, (k-1)-cut, so
Rank(AI’,O’) <k
O


   
Y1 Y2 Y3
Yn
Superconcentrators [Valiant]
Theorem: [Valiant]
Linear size SCn
I
Cn
Proof: [Pippenger]
(1)|I’|=|O’| n/2
recursion
(2) |I’|=|O’|> n/2
Matching & pigeonhole
Reduce to case (1)
Mn
SC2n/3
SCn
|E(SCn)|=
= |E(SC2n/3)|+2|E(Cn)|+n
= |E(SC2n/3)|+O(n) = O(n) O
Cn
Distributed routing
[Sh,PY,Up,ALM,AC…]
n inputs, n outputs, many disjoint paths
- Permutation networks
- Non-blocking network
- Wide-sense non-blocking networks
- on-line routable networks
- ….
Building block G - some expander
Theorem: [Alon-Capalbo]
Let G be a sufficiently strong expander.
Given (s1,t1),(s2,t2),…(sk,tk), k < n/(log n),
one can efficiently find (si,ti) edge-disjoint
paths between them, on-line!
G
Sorting networks [Ajtai-Komlos-Szemeredi]
n inputs (real numbers), n outputs (sorted)
Many sorting algorithms of O(n log n) comparisons
Many sorting networks of O(n log2 n) comparators
Thm: [AKS] Explicit network with O(n log n) comparators
Proof: Extremely sophisticated use & analysis of expanders
Cor: Monotone Boolean formula for majority
(derandomizing a probabilistic existence proof of Valiant)
Derandomization
Deterministic error reduction
G [2n,d, 1/8]-graph
G explicit!
Bx
Pr[error] < 1/3
|Bx|<2n/3
r1

x
Alg
{0,1}
random
strings
r

x
n
rk

Alg
x
Alg
Majority
Thm [Chernoff] r1 r2…. rk independent (kn random bits)
Thm [AKS] r1 r2…. rk random path (n+ O(k) random bits)
then Pr[error] = Pr[|{r1 r2…. rk }Bx}| > k/2] < exp(-k)
Metric embeddings
Metric embeddings (into l2)
Def: A metric space (X,d) embeds with distortion 
into l2 if  f : X  l2 such that for all x,y
d(x,y)   f(x)-f(y)    d(x,y)
Theorem: [Bourgain]
Every n-point metric space has a
O(log n) embedding into l2
Theorem: [Linial-London-Rabinovich] This is tight! Let (X,d)
be the distance metric of an [n,d]-expander G.
Proof: f,(AG-J/n)f   (G) f2
( 2ab = a2+b2-(a-b)2 )
(1-(G))Ex,y [(f(x)-f(y))2]  Ex~y [(f(x)-f(y))2] (Poincare
(clog n)2
All pairs
Neighbors
2
inequality)
Metric embeddings (into l2)
Def: A metric space (X,d) has a coarse embedding into l2 if
 f : X  l2 and increasing, unbounded functions ,:RR
such that for all x,y
(d(x,y))   f(x)-f(y) 2  (d(x,y))
Theorem: [Gromov] There exists a finitely generated,
finitely presented group, whose Cayley graph metric has no
coarse embedding into l2
Proof: Uses an infinite sequence of Cayley expanders…
Comment: Relevant to the Novikov & Baum-Connes
conjectures
Nonlinear spectral gaps and
metric embeddings into convex spaces
Poincare inequality: G any expander.   f : V  l2
Ex,y [f(x)-f(y)2]   Ex~y [f(x)-f(y) 2]
 = 1/(1-(G))
Theorem: [Matousek] G any expander.  p f : V  lp
p
Ex,y [f(x)-f(y) p ]
 pEx~y [f(x)-f(y)
p
P ]
Theorem: [Lafforgue, Mendel-Naor] Construct explicit
G (super-expander) K  K f : V  lp
2
2
Ex,y [f(x)-f(y) K]  K Ex~y [f(x)-f(y)  K ]
Theorem: [Kasparov-Yu, Gromov] Such family of constant
degree expanders give rise to a metric space X with no
coarse embedding into any uniformly convex space.
Constructions
Expansion of Finite Groups
G finite group, SG, symmetric. The Cayley graph
Cay(G;S) has xsx for all xG, sS.
Cay(Cn ; {-1,1})
Cay(F2n ; {e1,e2,…,en})
(G)  1-1/n2
(G)  1-1/n
Basic Q: for which G,S is Cay(G;S) expanding ?
Algebraic explicit constructions
[Margulis,GaberGalil,Alon-Milman,Lubotzky-Philips-Sarnak,…Nikolov,Kassabov,..]
A = SL2(p) : group 2 x 2 matrices of det 1 over Zp.
S = { M1 , M2 } : M1 = ( 10 11 ) , M2 = ( 11 01 )
Theorem. [LPS] Cay(A,S) is an expander family.
Proof: “The mother group approach”:
Appeals to a property of SL2(Z) [Selberg’s 3/16 thm]
Strongly explicit: Say that we need n bits to
describe a matrix M in SL2(p) . |V|=exp(n)
Computing the 4 neighbors of M requires poly(n) time!
Algebraic Constructions (cont.)
Very explicit
-- computing neighbourhoods in logspace
Gives optimal results Gn family of [n,d]-graphs
-- Theorem. [AB]
d(Gn)  2 (d-1)
--Theorem. [LPS,M] Explicit d(Gn)  2 (d-1)
(Ramanujan graphs)
Recent results:
-- Theorem [KLN] All* finite simple groups expand.
-- Theorem [H,BG] SL2(p) expands with most generators.
-- Theorem [BGT] same for all Chevalley groups
Zigzag graph product
Combinatorial construction
of expanders
Explicit Constructions (Combinatorial)
-Zigzag Product [Reingold-Vadhan-W]
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
Step in cloud
Step between clouds
Step In cloud
Thm. [RVW]
G z H is an [nm,d2,+]-graph,
G z H is an expander iff G and H are.
Combinatorial construction of expanders.
H
Proof of the zigzag theorem
Proof: Information theoretic view of expanders.
When is G an expander? Random walk is entropy boost!
p, p’ distributions on V before and after a random step.
“Def”:G is an expander iff whenever Ent(v) << log n,
Ent(v’) > Ent(v) + 
p’
(>0)
v’
Ent0(p) = log |supp(p)|
Ent1(p) = Shannon’s ent.
Ent2(p) = log
p2
v
p
Proof of the zigzag thm [Reingold-Vadhan-W]
Thm. [RVW] G, H expanders, then so is G z H
(u,d)
(v,b)
v-cloud
v
(v,a)
Want: Ent(u,d) > Ent(v,a) + 
(u,c)
u
H
u-cloud
(assuming Ent(v,a) << log nm)
Case 1: Ent(a|v) << log m  Ent(b|v) > Ent(a|v) +  
Mutually
Ent(u,d)  Ent(v,b)  Ent(v,a) + 
exclusive?
Case 2: Ent(a|v)= log m  Ent(b|v)= log m  Ent(v) << log n
Linear
Algebra!
 Ent(u)  Ent(v) +   Ent(c|u) << log m
 Ent(u,d)  Ent(u,c) + 
Iterative Construction of Expanders
G an [n,m,]-graph. H an [m,d,] -graph.
Theorem. [RVW] G z H is an [nm,d2,+]-graph.
The construction:
Start with a constant size H a [d4,d,1/4]-graph.
• G1 = H
2
• Gk+1 = Gk2 z H
Weakly explicit construction.
Strongly: (Gk  Gk)2 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.
Consequences of the zigzag product
- Isoperimetric inequalities beating e-value bounds
[Reingold-Vadhan-W, Capalbo-Reingold-Vadhan-W]
- Connection with semi-direct product in groups
[Alon-Lubotzky-W]
- New expanding Cayley graphs for non-simple groups
[Meshulam-W] : Iterated group algebras
[Rozenman-Shalev-W] : Iterated wreath products
- SL=L : Escaping every maze deterministically [Reingold ’05]
- Super-expanders [Mendel-Naor]
- Monotone expanders [Dvir-W]
SL=L: escaping mazes and
navigating unknown terrains
Getting out of mazes / Navigating unknown
terrains (without map & memory)
n–vertex maze/graph Only a local view (logspace)
Theseus
Crete
1000BC
Mars
2006
Ariadne
Thm [Aleliunas-KarpLipton-Lovasz-Rackoff ’80]
A random walk will visit
every vertex in n2 steps
(with probability >99% )
Thm [Reingold ‘05] : SL=L:
A deterministic walk,
computable in Logspace,
will visit every vertex.
Uses ZigZag expanders
Expander from any connected graph [Reingold]
Analogy with the iterative construction
G an [n,m,]-graph.
G an [n,m, 1-]-graph.
H an [m,d,] -graph.
H an [m,d,1/4] -graph.
Theorem. G z H is an [nm,d2,+]-graph.
The construction:
Fix a constant size H a [d4,d,1/4]-graph.
[nm,d2, 1-/2]-graph.
H a [d10,d,1/4]-graph.
• G1 = H 2
• G1 = G
• Gk+1 = Gk2 z H
• Gk+1 = Gk5 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.
Theorem [R]
G1 is [n, d2, 1-1/n3] 
Gclog n is [nO(1), d2, ½]
Undirected connectivity in Logspace [R]
Algorithm
- Input G=G1 an [n,d2]-graph
- Compute Gclog n
-Try all paths of length clog n from vertex 1.
Correctness
- Gi+1 is connected iff Gi is
- If G is connected than it is an [n,d2, 1-1/n3]-graph
- G1 connected  Gclog n has diameter < clog n
Space bound
- Gi Gi+1 in constant space (squaring & zigzag are local)
- Gclog n from G1 requires O(log n) space
Zigzag graph product &
Semi-direct group product
Expansion in non-simple groups
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)
Connection: semi-direct product is a special case of zigzag
Assume <T> = B, <S> = A , S = sB (S is a single B-orbit)
Theorem [ALW] Cay(A  B, TsT ) = Cay(A,S ) z Cay(B,T )
Proof: By inspection (a,b)(1,t) = (a,bt)
(Step in a cloud)
(a,b)(s,1) = (asb,b) (Step between clouds)
Theorem [ALW] Expansion is not a group property
Theorem [MW,RSW] Iterative construction of Cayley expanders
Example:
z
=
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 =Cay(A,S), H = Cay(B,T), and
G z H = Cay(A  B, {1 }e1 {1 } )
Is expansion a group property?
[Lubotzky-Weiss’93] Is there a group G, and two
generating subsets |S1|,|S2|=O(1) such that
Cay(G;S1) expands but Cay(G;S2) doesn’t ?
(call such G schizophrenic)
nonEx1: Cn
- no S expands
nonEx2: SL2(p)-every S expands[Breuillard-Gamburd’09]
[Alon-Lubotzky-W’01] SL2(p)(F2)p+1 schizophrenic
[Kassabov’05]
Symn
schizophrenic
Expansion in Near-Abelian Groups
G group. [G;G] commutator subgroup of G
[G;G] = <{ xyx-1y-1 : x,y G }>
G= G0 > G1> … > Gk = Gk+1
Gi+1=[Gi;Gi]
G is k-step solvable if Gk=1.
Abelian groups are 1-step solvable
loglog….log
k times
[Lubotzky-Weiss’93] If G is k-step solvable,
Cay(G;S) expanding, then |S| ≥ O(log(k)|G|)
[Meshulam-W’04] There exists k-step solvable Gk,
|Sk| ≤ O(log(k/2)|Gk|), and Cay(Gk;Sk) expanding.
Near-constant degree expanders for
near Abelian groups [Meshulam-W’04]
Iterate: G’= G  FqG
Start with G1 = Z2
Get G1 , G2,…, Gk ,…
|Gk+1|>exp (|Gk|)
S1 , S2,…, Sk ,…
<Sk > = Gk |Sk+1|<poly (|Sk|)
- |Sk|  O(log(k/2)|Gk|)
- Cay(Gk, Sk)
deg “approaching” constant
expanding
Theorem/Conjecture: Cay(G,S) expands, then G has
at most exp(d) irreducible reps of dimension d.
Expanders from iterated wreath
products [Rozenman-Shalev-W’04]
d fixed.
Gk = Aut*(Tdk)
Odd automorphisms of a depth-k
uniform d-ary tree
1
0
d=3, k=2
i  A3
2
3
Iterative: Gk+1 = Gk  Ad
Thm: Gk expands with explicit O(1) generators
Ingredients: zigzag, equations in perfect
groups[Nikolov], correlated random walks expand,…
Beating eigenvalue
expansion
Beating e-value expansion [WZ, RVW]
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!
Applications
Sorting & selection in rounds, Superconcentrators,…
Lossless expanders
[Capalbo-Reingold-Vadhan-W]
Task: Construct an [n,d]-graph in which every set S,
|S|<<n/d has > c|S| neighbors. Max c (vertex expansion)
Upper bound: cd
Ramanujan graphs: [Kahale] 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
Error Correcting Codes [Shannon, Hamming]
C: {0,1}k  {0,1}n
Rate (C) = k/n
C=Im(C)
Dist (C) = min dH(C(x),C(y))
C good if Rate (C) = (1), Dist (C) = (n)
Theorem: [Shannon ‘48] Good codes exist (prob. method)
Challenge: Find good, explicit, efficient codes.
- Many explicit algebraic constructions: [Hamming, BCH,
Reed-Solomon, Reed-muller, Goppa,…]
- Combinatorial constructions [Gallager, Tanner, LubyMitzenmacher-Shokrollahi-Spielman, Sipser-Spielman..]
Thm: [Spielman] good, explicit, O(n) encoding & decoding
Inspiration: Superconcentrator construction!
Graph-based Codes [Gallager’60s]
C: {0,1}k  {0,1}n
Rate (C) = k/n
C=Im(C)
Dist (C) = min dH(C(x),C(y))
C good if Rate (C) = (1), Dist (C) = (n)
0
+
n-k
n
1
1
zC iff Pz=0
0
0
+
0
+
0
+
1
0
+
0
Pz
+
0
G
1
1
z
C is a linear code
LDPC: Low Density Parity Check
Trivial
0
(G has constant degree)
Rate (C)  k/n , Encoding time = O(n2)
G lossless  Dist (C) = (n), Decoding time = O(n)
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
Pw
1
+
0
1
1
w
Decoding algorithm [Sipser-Spielman]: while Pw0 flip all
wi with i  FLIP = { i : (i) has more 1’s than 0’s }
B = 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
Rapidly mixing
Markov chains:
Uniform generation and
enumeration problems
Expansion of (exponential-size) graphs which
arise naturally (as opposed to specially designed)
Volumes of convex bodies
Given (implicitly) a convex body K in Rd (d large!)
(e.g. by a set of linear inequalities)
Estimate volume (K)
Comment: Computing volume(K) exactly is #P-complete
Algorithm: [Dyer-Frieze-Kannan ‘91]:
Approx counting  random sampling
Random walk inside K.
Rapidly mixing Markov chain.
Techniques: Spectral gap 
isoperimetric inequality
K
Statistical mechanics
Example: the dimer problem
Count # of domino configurations?
approximately
Given G, count the number of
v  HG
perfect matchings
G
Glauber Dynamics (MCMC sampling the Gibbs distribution)
Construct HG on configurations, with edges representing
local changes (e.g. rotate adjacent parallel dominos).
Then run a random walk for “sufficiently long” time.
Theorems:[Jerrum-Sinclair] poly(n) time convergence for
-near-uniform perfect matching in dense & random graphs
-sampling Gibbs dist in ferromagnetic Ising model
Techniques: coupling, conductance,…
Generating random group elements
Given: S={g1,g2,…gd} generators of a group G (of size n)
Find: a near-uniform element x of G (gG: Pr[x=g] < /n )
Straight-line program (SLP) [Babai-Szemeredi]:
x1,x2,…xt where every xi is either:
(1)A generator gj or gj-1
Techniques:
local expansion,
(2) xkxm or xm-1 for m,k < i
Arithmetic combinatorics
Theorem:[BS] G,S,g  an SLP for g of length t=O(log2 n)
Theorem:[Babai] G,S  a near-uniform SLP* of t=O(log5 n)
Theorem:[Cooperman,Dixon,Green] …. SLP* of t=O(log2 n)
Proof: Cube(z1,z2,…zt) = {subwords of zt-1…z2-1z1-1 z1z2…zt )
x1,x2,…xr with xk+1 R Cube(x1,x2,…xk), with r=O(log n)
Extensions of expanders
Dimension Expanders
Expander
Permutations 1, 2, …, k: [n]  [n] are an expander if
for every subset S[n], |S| < n/2
 i[k] s.t. |TiSS| < (1-)|S|
Dimension expander [Barak-Impagliazzo-Shpilka-W’01]
Linear operators T1,T2, …,Tk: Fn  Fn are (n,F)dimension expander if subspace VFn with dim(V) < n/2
 i[k] s.t. dim(TiVV) < (1-) dim(V)
Fact: k=O(1) random Ti’s suffice for every F,n.
[Lubotzky-Zelmanov’04] Construction for F=C
What about finite fields?
Monotone Expanders
f: [n]  [n] partial monotone map:
x<y and f(x),f(y) defined, then f(x)<f(y).
f1,f2, …,fk: [n]  [n] are a k-monotone expander
if fi partial monotone and the (undirected) graph on [n]
with edges (x,fi(x)) for all x,i, is an expander.
[Dvir-Shpilka] k-monotone exp  2k-dimension exp F,d
Explicit (log n)-monotone expander
[Dvir-W’09] Explicit (log*n)-monotone expander (zig-zag)
[Bourgain’09] Explicit O(1)-monotone expander
[Dvir-W’09] Existence  Explicit reduction
Open: Prove that O(1)-mon exp exist!
Real Monotone Expanders [Bourgain’09]
Explicitly constructs
f1,f2, …,fk: [0,1]  [0,1] continuous, Lipshitz, monotone
maps, such that for every S  [0,1] with (S)< ½,
there exists i[k] such that (Sf2(S)) < (1-) (S)
Monotone expanders on [n] – by discretization
M=( ac db )SL2(R), xR, let fM(x) = (ax+b)/(cx+d)
Take ’-net of such Mi’s in an ’’-ball around I.
Proof:
-Expansion in SU(2) [Bourgain-Gamburd]
-Tits alternative [Bruillard]
-…
Open Problems
Open Problems
Characterize: Cayley expanders
Construct: Lossless expanders
Construct:
Rate concentrators
of constant
left degree |X| N
N3
N2
|Γ(X)||X|