An Elementary Construction of Constant-Degree Expanders Noga Alon , Oded Schwartz and Asaf Shapira ** *Tel-Aviv University, Israel **Microsoft Research,

An Elementary Construction of Constant-Degree

Expanders

Noga Alon*, Oded Schwartz* and Asaf Shapira**

*Tel-Aviv University, Israel**Microsoft Research, USA

SODA 2007

2

Expanders

Sparse Highly connected

[n,d,]-expandern vertices, d regular

For every |S| ≤ n/2 d|S| ≤ |E(S,V \ S)|

\V SS

3

Applications of Expanders

Robust networks Derandomization Error Correcting Codes PCP Hardness of approximations …

4

Eigenvalues of Graphs

G: [n, d, ]-expander

Adjacency matrix A: Ai,j = #(i,j) edges

Eigenvalues1 ¸ 2 ¸ … ¸ n

1 = d, v1=1n

= maxi 1{|i|}

5

Expansion vs. Eigenvalues

G: [n, d, ]-expander

iff < d

Thm: [Alon-Milman84, Dodziuk84, Alon86]

(Actually 2)

12 (1 ) 2(1 )d d

6

Existence

[Pinsker73]:

9 > 0, s. t. 8 d ¸ 3 and (even) n,

9 [n, d, ]-expander.

7

Constructions Explicit:

Construct G

» E.g, for reductions» Time = Poly(n)

» or even Space = log n

» Which n’s? – 8n.

Fully Explicit: Find the i’th neighbor of v

» E.g, for derandomization» Time = Poly(|v|) = Poly(log n)» or even Space = loglog n

» Which n’s? – |G|=Poly(n) usually suffices.

8

Some Previous Results

[Margulis73], [Gabber-Galil81]Algorithm for Constant degree expanders.

…[Lubotzky-Phillips-Sarnak88], [Margulis88]Ramanujan Graphs.

[Alon-Roichman94]Polylog degree.

[Reingold-Vadhan-Wigderson02]Iterative construction.

9

Our Contribution

Thm:

9 8 n:

Explicit [Poly(n), O(1), ]-expander» Simple to construct» Simple to analyze

Applying the above we obtain:

1. Fully Explicit [Poly(n),O(1), ]-expander

2. Explicit [n, O(1), ]-expander

10

G2

Replacement Product: G1 ® G2

G1 : [n, d1, δ1]-expander

G2 : [d1, d2, δ2]-expander

G1

For all v of G1:

1. Split

2. Install G2

3. Duplicate edges

11

G1 ® G2: Properties


G2 : [d1, d2, δ2]-expander G2 G1G1 ® G2 : [nd1, 2d2, δ(δ1,δ2)]-expander

12

G1 ® G2: Some History



[Gromov83] : Spectral Analysis

[Reingold-Vadhan-Wigderson02] : via Zig-Zag

[Papadimitriou-Yannakakis91] : for Inapprox.

[Dinur05] : for PCP

13

G1 ® G2: Expansion


G2 : [d1, d2, δ2]-expander G2 G1

Thm:G1 ® G2 is a [nd1, 2d2, δ1

2 ¢ δ2/80)]-expander

[here]: Simple combinatorial proof.

14

E1: [n, log2n, ¼ ]-expander

[Special case of Alon-Roichman]

E2: [log2n, (loglogn)2, ¼ ]-expander

[A&R again]

E3: [2(loglog n)2, 3, ]-expander

(Pinsker, exhaustive search)

Return E = (E1 ® E2) ® E3

New Construction (roughly)

Constant Degree Expander Algorithm

E: [Poly(n), 6, ’ ]-expander

Proof

Proof

15

G1 ® G2: Combinatorial Proof

G1: [n, d1, δ1]-expander

G1 ® G2: [nd1, 2d2, δ12 ¢ δ2/80)]-expander

G2: [d1, d2, δ2]-expander

S V \ S

G1 ® G2

16

G2

G2

G2

G2

G2

G2

G2

G2

G2


G2

G1

G2




G2X

G2?

G1 ® G2

17





G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G1

G2

G2X

G2?

+G2

G2

G2G2

--

X

G1 ® G2

18

A&R: Proof

(a special case of [Alon-Roichman94]).

8 r, q=2t, LD(q,r):

Vertices: (a0,…,ar) ai 2 F2t = [q]

Edges:(x,y) 2 [q]2

The (x,y) neighbor of (a0,…,ar) is:(a0,…,ar)+y¢(1,x,x2,…,xr).

19

LD(q,r): Properties

LD(q,r) :n = qr+1 verticesd = q2-regular

Thm: [Alon-Roichman94]: (LD(q,r)) · rq

For r = q/2 ¸ ½(1- / d) ¸ ¼ log2(n) ¸ (½q log q)2 > q2 = d

LD(q,r) : [n, O(log2n), ¼]-expander.

20

LD(q,r): Eigenvalues

M: n £ n matrix of LD(2t,r)

L : 2t! {0,1} L(x0,…xt-1)=x0

Eigenvectors:

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F

21


1. va are orthogonal

2. va are eigenvectors

(what eigenvalues?)

+

va are all eigenvectors

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F

22


1. va are orthogonal

Proof:

<va, va’> = b va(b) va’(b) W.l.o.g, a0 a’0.

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F

0 0

0

0 0 01

( ) ( ' )

'

( ' )

( ' ' )

1

1

1

r r

i i i i

r

i i i

r

i i i

L a b L a b

a a

L a a b

L a a b a a b

v b v b

All values

23


2. va are eigenvectors

Proof:

va (b + c) = va (b) va(c)

(M va)(b) = c 2 Fr+1 Mbc¢ va(c)

= x,y 2 F va(b+y(1,x,...,xr))

= (x,y 2 F va(y,yx,...,yxr))¢ va(b)

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F F

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P

x;y2F va(y; yx; :::; yxr)

va(b)(Mva) (b) =

Xc2F r+1

Mbc va(c) =Xx;y2F

va(b+y(1;x; :::;xr)) =

0@Xx;y2F

va(y; yx; :::; yxr)

1A va(b) :

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P


va(b)

The (x,y) neighbor of

b

a

24


2. va are eigenvectors!

What are a ?a = x,y 2 F va((y,yx,...,yxr))

pa(x) = ir=0 aixi

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F F

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P


va(b)(Mva) (b) =

Xc2F r+1

Mbc va(c) =Xx;y2F

va(b+y(1;x; :::;xr)) =

0@Xx;y2F

va(y; yx; :::; yxr)

1A va(b) :

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P


va(b)

( )

,

1 a iL y p x

ax y F

b

( ) ( )

, : 0 , : 0

1 1a a

a a

L y p x L y p x

x y F p x x y F p x

25


2. va are eigenvectors.

What are a ?

0

( )

0 2

1

,..., ,

r

i i

t

L a b

a

r i

v b

a a a a F F

(Mva) (b) =X

c2F r+1Mbc va(c) =

Xx;y2F

va(b+y(1;x; :::;xr)) =

0@Xx;y2F

va(y; yx; :::; yxr)

1A va(b) :

( ) ( )

, : 0 , : 0

1 1a a

a a

L y p x L y p x

ax y F p x x y F p x

=1, 8 y

= |F| = q

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P


va(b)

(Mva) (b) =P

c2F r+1Mbc va(c) =P

x;y2F va(b+y(1;x; :::;xr)) =P


va(b)

all values

= 0

a = (0,…0) ) a = q2

a (0,…0), ) pa has at most r roots ) a · rq

26

Variants

» For every n» Fully explicit

27

For Every n

LD(q,r) :

n = qr+1 vertices

d = q2-regular

E1: [q2,3, ]-expander Search

E2: [q6,q2,1/4]-expander LD(q,5)

E3: [q4(r+1),q8,1/4]-expander LD(q4,r) q4/100≤ r ≤ q4/2

E4 = E3 ® (E2 ® E1)

E4: [q4r+12,12, ’]-expander

28

Poly(n)

E4: [q4r+12,12, ’]-expander q4/100≤ r ≤ q4/2

q = 2t

n = q^ (q4) q4 = (lg n / lg lg n)

t’ = t + 1 q’ = 2q, r’ = 16rn’ ≤ (2q)4r’+12 = Poly(n)

r’ = r - 1 n’ = n/ q4

q 4(1/2 q^4)+12 > (2q)4(16 q^4 /100) +12

n, n lglgn / lgn ≤ n0 ≤ n

29

(n)

n, n lglgn / lgn ≤ n0 ≤ n

a = n / n0 < lgn / lglgn

if a < 100 – done. E5 : [12a, 3, ]-expander search: (lgn/lglgn) (lg n / lglg n) = Poly(n) Duplicate edges

E = E4 ® E5

E:[12n, 6, ’’]-expander

30

Fully Explicit

Vertex naming: (x,y) The i’th neighbor of (x,y)

» If i ≤ d2 then (x, y’) : y’ is neighbor i in G2

» else (x’, y): x’ is neighbor y in G1

G2 G1


G2 : [d1, d2, δ2]-expander

G1 ® G2 : [nd1, 2d2, δ(δ1,δ2)]-expander

31

Connection Scheme

G2 is d2-edge-colorable+

G1 ® G2 is 2d2-edge-colorable.



Connection scheme? (rotation map)

e.g: d1-edge-coloring for G1.

32

Fully Explicit

G1 ® G2

» Fully Explicit» D-edge-colorable

Alon-Roichman:» Fully Explicit» D-edge-colorable

Pinsker» Fully Explicit» D-edge-colorable

33

A&R: Edge Colorability


8 r, q=2t, LD(q,r):


Edges: (x,y) 2 [q]2 the (x,y) neighbor of (a0,…,ar):

(a0,…,ar)+y¢(1,x,x2,…,xr).

34

Conclusions and Open Problems

A (Fully) explicit constant degree expander» Simple to construct» Simple to analyze

Exploiting the simplicity? Simple combinatorial proof

» for [n, poly-log n, ¼ ]-expander?» for graph powering?» for (near) Ramanujan graphs?

35

Expanders Construction

Thank You.

36

G2

G2

G2

G2

G2

G2

G2

G2

G2


G2

G1

G2




37





(1-1/4)d1Each contributes:

1 d1 /4 ¢ 2 d2

If (small) ¸ 1|S|/10

#sets ¸ 1|S|/10d1

Total:

1d1/4¢2d2¢ 1|S|/10d1

= 122/80 ¢ 2d2¢|S|

Else

small) < 1|S|/10 (Large)¸ (1-1/10) |S|

38





Else

small) < 1|S|/10 (Large)¸ (1-1/10) |S|

1

1

1 111 1

11210 2#3

1 14 4

S ndSlarge sets n

dd d

39





1

1

1

small<10

21 #

10 3

S

Slarge sets n

d

G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G1

G2

40


G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G2




1

1

1

small<10

21 #

10 3

S

Slarge sets n

d

G2 G2 G2G2

G1:Total ¸ ½ 1d1#large-sets

G1 ® G2: ½1d1d2#large-sets - corrections

41


G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G2




1

1

1

small<10

21 #

10 3

S

Slarge sets n

d

G2 G2 G2G2

At most:

¼ 1d1d2 #large-sets

42


G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G2




1

1

1

small<10

21 #

10 3

S

Slarge sets n

d

At most:

d2 (small) < d21|S| /10

· 1d2/10 ¢ #large-sets d1 10/(10-1)

· 1/9 1d1d2#large-sets

G2 G2 G2G2

43


1 1 2 11

1

1

21

1

2

2

1 1 1#

11

1

16

5 5 92

36 36 1

2 9

00

4

d d

d d large sets

d S dd

SS

G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G2




G2 G2 G2G2

¼ 1d1d2 #large-sets

½ 1d1d2#large-sets - corrections

1/9 1d1d2#large-sets

1

1

21 #

10 3

Slarge sets n

d

44

Compared to Random Walk

G2

G2

G2

G2

G2

G2

G2

G2

G2 G2

G1

G2




45

[n,polylog n,]-expander


8 r, q=2t, LD(q,r):


Edges: (x,y) 2 [q]2 the (x,y) neighbor of (a0,…,ar):

(a0,…,ar)+y¢(1,x,x2,…,xr).

46

LD(q,r): Properties

LD(q,r):n = qr+1 verticesd = q2-regularq2-edge-colorable

Thm: [Alon-Roichman94]: (LD(q,r)) · rq

Recall ½(1- / d) · If r · q/2 then LD(q,r) is a [qr+1, q2, ¼]-expander.

Documents

An Elementary Construction of Constant-Degree Expanders Noga Alon *, Oded Schwartz * and Asaf Shapira ** *Tel-Aviv University, Israel **Microsoft Research,

An Elementary Construction of Constant-Degree Expanders Noga Alon , Oded Schwartz and Asaf Shapira ** *Tel-Aviv University, Israel **Microsoft Research,