Expander graphs – applications and combinatorial constructions

Avi WigdersonIAS, 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

• Data Structures

• …

• Computational Information

• Approximate Counting

• Communication & Sorting Networks

Applications of Expanders

In Pure Math

• Graph Theory - …

• Group Theory – generating random group elements [Babai,Lubotzky-Pak]

• Number Theory Thin Sets [Ajtai-Iwaniec-Komlos-Pintz-Szemeredi] -Sieve method [Bourgain-Gamburd-Sarnak]

- Distribution of integer points on spheres [Venkatesh]

• Measure Theory – Ruziewicz Problem [Drinfeld, Lubotzky-Phillips-Sarnak], F-spaces [Kalton-Rogers]

• Topology – expanding manifolds [Brooks]

- Baum-Connes Conjecture [Gromov]

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

• Combinatorial: no small cuts, high connectivity

• Geometric: high isoperimetry

G(V,E)

V vertices, E edges

|V|=n ( ∞ )

d-regular (d fixed)

d

S |S|< n/2

|E(S,Sc)| > α|S|d (what we expect in a random graph)

α constant

S

Expanding Graphs - Properties

• Probabilistic: rapid convergence of random walk

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)

Expanding Graphs - Properties

• Algebraic: small second eigenvalue

G(V,E) V

V AG

AG(u,v) =

normalized adjacency matrix

(random walk matrix)

0 (u,v) E1/d (u,v) E

1 = 1 ≥ 2 ≥ … ≥ n ≥ -1

(G) = maxi>1 |i| =

max { AG v : v =1, vu }

(G) ≤ < 1

1-(G) “spectral gap”

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 AG

AG(u,v) =0 (u,v) E1/d (u,v) E

1 = 1 ≥ 2 ≥ … ≥ n eigenvalues

u = v1 v2 … vn eigenvectors

p: any probability distribution on V

p(t)= (AG)tp: distribution after t steps of a random walk

p(t) – u1 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 AG

AG(u,v) =0 (u,v) E1/d (u,v) E

1 = 1 ≥ 2 ≥ … ≥ n eigenvalues

u = v1 v2 … vn eigenvectors

S V 1S = i ivi 1 = |S|/n

T V 1T = i ivi 1 = |T|/n

1SAG1T = i iii = |S||T|/n + i>1 iiI

||E(S,T)|- d|S||T|/n | d(G)i>1ii d(G) |S||T| d(G)n

Expected cut size:like a random graph AG J/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 ( ) (s/n)3s < 1/2

Challenge: Explicit (small degree) expanders!

n s

Explicit constructionG(V,E) V

V AG

AG(u,v) =0 (u,v) E1/d (u,v) E{Gi(Vi,Ei)} infinite sequence of graphs

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

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 i

2 < 1+(n-1)(G)2

1/2d < (G)

V

V AG

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 > 2s 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]

1

0 1

100

Reliable circuits from unreliable components [von Neumann]

X2 X3

f

VV

VV

V

X1

V

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.

Possible? With small s’?

Reliable circuits from unreliable components [von Neumann]

X2 X3

f

VV

V

V

V

X1

V

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.

Possible? With small s’?

- Add Identity gatesII I

I

I

I

1

Reliable circuits from unreliable components [von Neumann]

f

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.

Possible? With small s’?

-Add Identity gates

-Replicate circuit

-Reduce errors

X2 X3

VV

V

V

V

X1

V

II I

I

I

I

X2 X3

VV

V

V

V

X1

V

II I

I

I

I

X2 X3

VV

V

V

V

X1

V

II I

I

I

I

X2 X3

VV

V

V

V

X1

V

II I

I

I

I

Reliable circuits from unreliable components[von Neumann, Dobrushin-Ortyukov, Pippenger]

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.

Possible? With small s’?

Majority “expanders”

of size O(log s)

Analysis:

Infection

Process 2

1

f

X2 X3

VV

V

V

V

X1

V

M

X2 X3

VV

V

V

V

X1

V

M

X2 X3

VV

V

V

V

X1

V

M

X2 X3

VV

V

V

V

X1

V

M

MMMM MMMM

MMMM MMMMMMMM

Bipartite expanders

(unbalanced) Bipartite expanders

G(V,E)

V’

V’’

u

w

u’ w’

u” w”

V’’’

H(V’,V’’’;E*) |V’|=n, |V’’’|=(2/3)n, degree=4dSV’ |S| n/2, |NV’’’ (S)| |S| S has a perfect matching to V’’’

SV’ |S| n/2, |NV’’ (S)| (3/2)|S|

Concentrators Cn [Bassalygo,Pinsker]

Cn

SV’ |S| n/2, S has a perfect matching to V’’

V’

V’’

n

2n/3

Networks

- Fault-tolerance- Routing- Distributed computing- Sorting

Superconcentrators [Valiant]

|I|=|O|=n

k n SCn I’ I O’ O|I’|=|O’|=kThere are k vertex disjoint paths I’ to O’

I

O

How many edges are needed for SCn ?

[V] This number is a Circuit lower bound forDisc. Fourier Transform

Superconcentrators & circuits [Valiant]

[V] |E(SCn)| is a circuit lower bound forlinear circuits computingany matrix A with allminors nonsingular y = Ax

k n I’ I O’ O|I’|=|O’|=k there are k Vertex disjoint paths I’ to O’

Otherwise, by Menger’sTheorem, (k-1)-cut, soRank(AI’,O’) <k

I

O

X1 X2 X3 Xn

Y1 Y2 Y3 Yn

Superconcentrators [Valiant]

Theorem: [Valiant] Linear size SCn

Proof: [Pippenger]

•|I’|=|O’| n/2 recursion(2) |I’|=|O’|> n/2Matching & pigeonholeReduce to case (1)

|E(SCn)|== |E(SC2n/3)|+2|E(Cn)|+n= |E(SC2n/3)|+O(n) = O(n)

I

O

SCn

SC2n/3

Cn

Cn

Mn

Distributed routing [Sh,PY,Up,ALM,AC…]

Gn 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-disjointpaths between them, on-line!

Sorting networks [Ajtai-Komlos-Szemeredi]

n inputs (real numbers), n outputs (sorted)

Many sorting algorithms of O(n log n) comparisonsMany sorting networks of O(n log2 n) comparators

Thm: [AKS] Explicit network with O(n log n) comparatorsProof: Extremely sophisticated use & analysis of expanders

Cor: Monotone Boolean formula for majority(derandomizing a probabilistic existence proof of Valiant)

Derandomization

Deterministic error reduction

Algx

r

{0,1}n

random

strings

Thm [Chernoff] r1 r2…. rk independent (kn random bits) Thm [AKS] r1 r2…. rk random path (n+ O(k) random bits)

Algx

rk

Algx

r1

Majority

G [2n,d, 1/8]-graphG explicit! Bx

Pr[error] < 1/3

then Pr[error] = Pr[|{r1 r2…. rk }Bx}| > k/2] < exp(-k)

|Bx|<2n/3

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) f2 ( 2ab = a2+b2-(a-b)2 )

(1-(G))Ex,y [(f(x)-f(y))2] Ex~y [(f(x)-f(y))2] (Poincare

inequality)

(clog n)2

2

All pairs Neighbors

Metric embeddings (into l2)

Def: A metric space (X,d) has a coarse embedding into l2 if f : X l2 and increasing, unbounded functions ,:RR 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

spacesPoincare 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

Ex,y [f(x)-f(y) ] pEx~y [f(x)-f(y) ]

Theorem: [Lafforgue, Mendel-Naor] Construct explicit

G (super-expander) K K f : V lp

Ex,y [f(x)-f(y) ] K Ex~y [f(x)-f(y) ]

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.

p p

p P

2 K

2 K

Constructions

Expansion of Finite GroupsG finite group, SG, symmetric. The Cayley graphCay(G;S) has xsx for all xG, sS.

Cay(Cn ; {-1,1}) Cay(F2n ; {e1,e2,

…,en})

(G) 1-1/n2 (G) 1-1/nBasic Q: for which G,S is Cay(G;S) expanding ?

Algebraic explicit constructions [Margulis,Gaber-Galil,Alon-Milman,Lubotzky-Philips-Sarnak,…Nikolov,Kassabov,..]

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!

A = SL2(p) : group 2 x 2 matrices of det 1 over Zp.

S = { M1 , M2 } : M1 = ( ) , M2 = ( ) 1 10 1

1 01 1

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.

Combinatorial construction of expanders.

H

v-cloud

Edges

Step in cloud

Step between clouds

Step In cloud

v uu-cloud

(v,k)

Thm. [RVW] G z H is an [nm,d2,+]-graph,

Definition. G z H has vertices {(v,k) : vG, kH}.

G z H is an expander iff G and H are.

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) + (>0)

Ent0(p) = log |supp(p)|

Ent1(p) = Shannon’s ent.

Ent2(p) = log p2

p

v

v’

p’

Proof of the zigzag thm [Reingold-Vadhan-W]

H

v-cloud v u

u-cloud

Thm. [RVW] G, H expanders, then so is G z H

(u,c)

(u,d)

Want: Ent(u,d) > Ent(v,a) + (assuming Ent(v,a) << log nm)

Case 1: Ent(a|v) << log m Ent(b|v) > Ent(a|v) +

Ent(u,d) Ent(v,b) Ent(v,a) +

Case 2: Ent(a|v)= log m Ent(b|v)= log m Ent(v) << log n

Ent(u) Ent(v) + Ent(c|u) << log m

Ent(u,d) Ent(u,c) +

(v,b)

(v,a)

Mutuallyexclusive?

LinearAlgebra!

Iterative Construction of Expanders

G an [n,m,]-graph. H an [m,d,] -graph.

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.

Gk+1 is a [d 4(k+1), d 2, ½]-graph.

Theorem. [RVW] G z H is an [nm,d2,+]-graph.

• Gk+1 = Gk2 z H

Weakly explicit construction.Strongly: (Gk Gk)2 z H

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)

Theseus

Ariadne

Thm [Aleliunas-Karp-Lipton-Lovasz-Rackoff ’80] A random walk will visit every vertex in n2 steps (with probability >99% )

Only a local view (logspace)n–vertex maze/graph

Thm [Reingold ‘05] : SL=L:A deterministic walk, computable in Logspace, will visit every vertex. Uses ZigZag expanders

Crete1000BC

Mars2006

Expander from any connected graph [Reingold]

Analogy with the iterative construction

G an [n,m,]-graph.

H an [m,d,] -graph.

The construction:

Fix a constant size H a [d4,d,1/4]-graph.• G1 = H 2

Theorem. [RVW] Gk is a [d4k, d2, ½]-graph Proof: Gk

2 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. G z H is an [nm,d2,+]-graph.

• Gk+1 = Gk2 z H

G an [n,m, 1-]-graph.

H an [m,d,1/4] -graph. [nm,d2, 1-/2]-

graph.

H a [d10,d,1/4]-graph.• G1 = G

• Gk+1 = Gk5 z H

Gclog n is [nO(1), d2, ½]

Theorem [R] G1 is [n, d2, 1-1/n3]

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) : aA, bB}.

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)

Proof: By inspection (a,b)(1,t) = (a,bt) (Step in a cloud) (a,b)(s,1) = (asb,b) (Step between clouds)

Theorem [ALW] Cay(A B, TsT ) = Cay(A,S ) z Cay(B,T )

Theorem [ALW] Expansion is not a group property

Theorem [MW,RSW] Iterative construction of Cayley expanders

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 =Cay(A,S), H = Cay(B,T), and

G z H = Cay(A B, {1 }e1 {1 } )

z

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

[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.

loglog….log k times

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|)

- Cay(Gk, Sk) expanding

- |Sk| O(log(k/2)|Gk|) deg “approaching” constant

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

Iterative: Gk+1 = Gk Ad

Thm: Gk expands with explicit O(1) generators

Ingredients: zigzag, equations in perfect groups[Nikolov], correlated random walks expand,…

0

1 32

d=3, k=2i A3

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 dRamanujan 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: cd

Ramanujan graphs: [Kahale] 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

Error Correcting Codes [Shannon, Hamming]

C: {0,1}k {0,1}n C=Im(C)

Rate (C) = k/n 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, Luby-Mitzenmacher-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 C=Im(C)

Rate (C) = k/n Dist (C) = min dH(C(x),C(y))

C good if Rate (C) = (1), Dist (C) = (n)

zC iff Pz=0 C is a linear code

LDPC: Low Density Parity Check (G has constant degree)Trivial Rate (C) k/n , Encoding time = O(n2)

G lossless Dist (C) = (n), Decoding time = O(n)

n

n-k

1 1 0 1 0 0 1 1 z

0 0 0 0 0 0 Pz + + + + + +

G

Decoding

Thm [CRVW] Can explicitly construct graphs: k=n/2,

bottom deg = 10, B[n], |B| n/200, |(B)| 9|B|

B = corrupted positions (|B| n/200)

B’ = set of corrupted positions after flip

Decoding algorithm [Sipser-Spielman]: while Pw0 flip all wi with i 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

1 1 1 0 1 0 1 1 w

0 0 1 0 1 1 Pw + + + + + +

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

Algorithm: [Dyer-Frieze-Kannan ‘91]: Approx counting random sampling Random walk inside K. Rapidly mixing Markov chain. Techniques: Spectral gap isoperimetric inequality

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

K

Statistical mechanicsExample: the dimer problemCount # of domino configurations?

Given G, count the number of perfect matchings GGlauber 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 modelTechniques: coupling, conductance,…

v HG

approximately

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 (gG: Pr[x=g] < /n )

Straight-line program (SLP) [Babai-Szemeredi]:x1,x2,…xt where every xi is either:•A generator gj or gj

-1

• xkxm or xm-1 for m,k < i

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)

Techniques: local expansion, Arithmetic combinatorics

Extensions of expanders

Dimension ExpandersExpanderPermutations 1, 2, …, k: [n] [n] are an expander if for every subset S[n], |S| < n/2 i[k] s.t. |TiSS| < (1-)|S|

Dimension expander [Barak-Impagliazzo-Shpilka-W’01]Linear operators T1,T2, …,Tk: Fn Fn are (n,F)-dimension expander if subspace VFn with dim(V) < n/2 i[k] s.t. dim(TiVV) < (1-) dim(V)

Fact: k=O(1) random Ti’s suffice for every F,n.

[Lubotzky-Zelmanov’04] Construction for F=CWhat about finite fields?

Monotone Expandersf: [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 reductionOpen: Prove that O(1)-mon exp exist!

Real Monotone Expanders [Bourgain’09]

Explicitly constructsf1,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 (Sf2(S)) < (1-) (S)

Monotone expanders on [n] – by discretization

M=( )SL2(R), xR, 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] -…

a bc d

Open Problems

Open Problems

Characterize: Cayley expanders

Construct: Lossless expanders

Construct: Rate concentratorsof constant left degree

N3

N2

|X| N|Γ(X)||X|