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Randomness Conductors and Constant-Degree Expansion beyond the Degree / 2 Barrier

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### Randomness Conductors andConstant-Degree Expansionbeyond the Degree / 2 Barrier

OutlineOutline

Salil Vadhan - Harvard University

Michael Capalbo - DIMACS

Omer Reingold - Weizmann

Avi Wigderson - IAS

Slides by Guy Fridland

Outline

- Introduction & Overview
- Randomness conductors
- The Original Zig-Zag product & its limitations
- The New Zig-Zag product for conductors
- Explicit Construction

The big picture

Expanders. What are they good for?

- High Connectivity.
- No “bottle-neck”

Many applications!

- Network design.
- Sorting.
- Complexity theory

- Cryptography.
- Coding theory
- Proof complexity
- Many more…

D

Expander Graphs- Bipartite graph G with:
- N inputs, N outputs
- Every input connected to D
- outputs.
- Saw many definitions:
- Algebric (spectral gap)
- (1+d)-expander
- Definition : (K,A)-expander
- The graph is (K,A)-expander if every set X of at most K
- inputs is connected to at least A∙|X| outputs.

N

|X| K

|(X)| A |X|

N

|X| K

D

|(X)| A |X|

Expander GraphsGoals:

- Minimize the degree D
- Maximize the expansion A

Q: How large can A be?

- Trivial upper bound: A D.
- Random graphs: A=D-1.01
- But many applications need explicit (deterministic & efficient) constructions.
- Previously, best explicit expanders: A =D/2
- Algebric construction: optimal 2nd eigenvalue (Ramanujan graphs).

D

This Work: Constant-Degree “Lossless” ExpandersN

When:

A=(1-)∙D

0< small constant

we call the expander:

Lossless

|X| K

|(X)| (1-) D |X|

Why Bother with theDegree/2 Barrier?

- For most applications of expanders: the more expansion the better.
- Specific applications for lossless expanders:
- Distributed routing in networks [PU89,ALM96,BFU99].
- Expander codes [Gal63,Tan81,SS96,Spi96,LMSS01].
- “Bitprobe complexity” of storing subsets [BMRS00].
- Distributed storage schemes [UW87].
- Hard tautologies for various proof systems [BW99,ABRW00,AR01].

Outline

- Introduction & Overview
- Randomness conductors
- The Original Zig-Zag product & its limitations
- The New Zig-Zag product for conductors
- Explicit Construction

Measures of Expansion

- Set Expansion: No small cuts, high connectivity
- Expansion Factor:
- Algebraic: Small second eigenvalue
- Thm [...]: All equivalent.

Each measure has its limitations.

Extremely useful to adopt stronger measure of “randomness”

Min-Entropy ! [Zuckerman `90]

- Min-EntropyMeasures how much a dist. is close to uniform.
- Reminder:
- Def: Min-Entropy of a dist. X:

maximum are taken over

- Def: X is a k-source if H∞(X) ≥ k.
- i.e. the uniform dist. on a set of size 2k is a k-source.
- Def: Xis a (k,ε)-sourceif it ise-close to somek-source.
- From now on:
- N=2n, M=2m ,D=2d…

- Def: X, Ye-close if D(X,Y)≤ ε
- Measure of closeness: statistical difference: (D)

CON

d-bit seed

m-bit output

Randomness Conductors- General framework for“randomness enhancing” functions:
- Expanders, extractors,condensers, and theirrelatives…

All of the above may be viewed as

a function:

E : [N] x [D]→[M]

- Given guarantees on the randomness of the input dist. X.
- E gives guarantees on the randomness of the dist. E(X,UD)

A function Is a (kmax ,a,ε)-conductor, if for any 0≤k≤kmax, and any k-source X over {0,1}n,the dist. E(X,Ud) is a (k+a,ε)-source.

E : {0,1}n x {0,1}d → {0,1}m

n-bit input

“k amount of entropy”

CON

d-seed

m-bit output

e-close to dist. with ≥ k+a amount of entropy

Simple Conductor

- E gets 2 inputs:
- X: dist. With min-entropy k≤kmax
- The uniform dist. UD (min-entropy d)
- Output:
- e-close to dist. with at leastk+a amount of entropy

“m-a amount of entropy”

ExtCon

d-seed

m-bit output

e-close to dist. with ≥ m amount of entropy

Special Conductors:

Def: Extracting Conductor

A function Is a(a,ε)-extracting conductor if it is a (m-a,a,ε)-conductor.

E : {0,1}n x {0,1}d → {0,1}m

- Kmax = m-a
- If input entropy is m-a
- output e-close to uniform

a = d

input entropy is k

Seed entropy is d

output e-close to dist. With≥ k+d amout of entropy.

n-bit input

“k amount of entropy”

LossLessCon

d-seed

m-bit output

e-close to dist. with ≥ k+damount of entropy

Special Conductors:

Def: Lossless Conductor

A function Is a(kmax,ε)-Lossless conductor if it is a (kmax,a,ε)-conductor.

E : {0,1}n x {0,1}d → {0,1}m

- Next 2 conductors combine the above two cases(extracting & lossless)

Def: Buffer Conductor

A function Is a(a,ε)-buffer conductor if E is a (a,ε)-extracting conductor & is an (kmax,ε)-lossless conductor.

<E,C> : {0,1}n x {0,1}d → {0,1}m x {0,1}b

- Intuition:
- Think of a “Buffer conductor” as putting a bucket beneath a lossy conductor.
- So when we pour randomness (water) into it, the leftovers (unused randomness), are stored for later use.

d-bit seed

Conductor

Buffer

ExtractingConductor

LosslessConductor

m-bit output

b-bit seed

Buffer Conductor

The same as Buffer Conductor where n+d=m+b, & is a permutaion over {0,1}n+d

n-bit input

d-bit input

Conductor

- Is a permutation

Buffer

- is alossless conductor(lossless even if the seed is ‘bad’ because it is apermutation)

m-bit input

b-bit input

N

x

n-bit input

D=2d

(kmax,ε)Lossless

Conductor

d-bit

seed

M

y

(1-)DK

m-bit output

Lossless Conductors Vs Expanders- x{0,1}ny{0,1}mif LossCond(x,r)=y for some r {0,1}d
- (kmax,ε)-Lossless Conductor (2kmax,ε)-Lossless Expander

Explicit construction of a constant degreeLossless expander!

- Starting Point: Zig-Zag Graph Product [RVW00] Compose large graph w/ small graph to obtain a new graph which (roughly) inherits
- Size of large graph.
- Degree from the small graph.
- Expansion from both.
- Composition of Expanders via the Zig-Zag Product will imply constant degree.

Outline

- Introduction & Overview
- Randomness conductors
- The Original Zig-Zag product & its limitations
- The New Zig-Zag product for conductors
- Explicit Construction

Size: n*d1

Degree: d22

A step on Gi “adds ai bits of entropy”.

A step on G “adds min{a1,a2} bits of entropy

Suboptimal expansion!

Deficiency Can be traced back to the expander composition itself.

z

- G = G1 G2

- Expansion ≈ min{Expansion(G1), Expansion(G2)}

The Zig-Zag Product - Analysis

G1: (n,d1) G2: (d1,d2)

The Zig-Zag Product - Analysis

Lemma A:

Let (X1,X2) be a prob. on the vertices of the Zig-Zag product.

It suffices to consider only 2 extreme cases:

- H∞(X2 | X1 = x1) close to uniform
- H∞(X2 | X1 = x1) far from uniform

First step on small graph adds entropy.

Zig-Zag Analysis (Case I)- Case I: Conditional distributions within “clouds” far fr. uniform.

- Next two steps can’t lose entropy.

First small step does nothing.

- Step on big graph “scatters” among clouds (shifts entropy)

- Second small step adds entropy.

- Case II: Conditional distributions within clouds uniform.

- In each case, only one of two small steps adds entropy
- But paid for both in degree.
- Expansion is too low for lossless expanders
- The new Zig-Zag product for conductors manages to avoid this problem.

- Introduction & Overview
- Randomness conductors
- The Original Zig-Zag product & its limitations
- The New Zig-Zag product for conductors
- Explicit Construction

Zig-Zag for Conductors

n1

- Start with a lossy conductor E1
- Any constant degree expander will do.
- Goal: make it lossless.

d1

E1

n1

Zig-Zag for Conductors

n2

n1

- “Small” Conductor E2:
- Blow up each vertexto cloud of size 2n2

- Apply E2 to select edge for E1
- If input entropy large enough, output close to uniform
- If E2 input entropy too large,we lose entropy!

d2

E2

d1

E1

n1

Zig-Zag for Conductors

n2

n1

d2

E2

d1

E1

Idea:

Keep “buffers” to retain lost entropy

In lossy Conductors E1 & E2

n1

Zig-Zag for Conductors

- Let’s get formal:To define the product we need 3 objects:
- <E1,C1> : {0,1}n1 x {0,1}d1 → {0,1}m1 x {0,1}b1Permutation Conductor
- <E2,C2> : {0,1}n2 x {0,1}d2 → {0,1}d1 x {0,1}b2 Buffer Conductor
- E3 : {0,1}b1+b2 x {0,1}d3 → {0,1}m3 Lossless Conductor
- The Zig-Zag product for Conductors produces:

E : {0,1}n x {0,1}d → {0,1}m

n = n1+n2d = d2+d3m = m1+m3

Zig-Zag for Conductors

n2

n1

<E1,C1>: Permutation Conductor

d2

E2,C2

<E2,C2>: Buffer

Conductor

d1

E1,C1

<E3,C3>: Lossless

Conductor

b2

b1

d3

E3

m1

m3

Analysis of Entropy Flow

Case II:

Conditional

entropy within

clouds small.

n2

n1

d2

E2,C2

d1

E1,C1

b2

b1

d3

E3

n1

m3

- Introduction & Overview
- Randomness conductors
- The Original Zig-Zag product & its limitations
- The New Zig-Zag product for conductors
- Explicit Construction

Explicit Construction

- Fix : a = 1000log(1/e) d = 2a;
- (n-30a,6a,e)-Permutation Conductor
- <E1,C1> : {0,1}n-20a x {0,1}14a → {0,1}n-20 x {0,1}14a
- (14a,0,e)- Buffer Conductor
- <E2,C2> : {0,1}20a x {0,1}a → {0,1}14a x {0,1}21a
- (15a,e)-Lossless Conductor
- E1 : {0,1}35a x {0,1}a → {0,1}17a
- Claim: The resulting conductor
- E : {0,1}n x {0,1}2a → {0,1}n-3aIs an (n-30a,4e)-lossless conductor.

Explicit Construction

n2

n1

E1: (n-30a,6a,e)-Permutation

x2(20a)

x1(n-20a)

r2(a)

E2: (14a,0,e)- Buffer

E2,C2

E3: (15a,e)-Lossless

y2(14a)

E1,C1

C1: keep the whole seed (14a)

z2(21a)

z1(14a)

C2: keep the (small) input & seed(20a +a = 21a)

E3

y3(17a)

y1(n-20a)

E: (n-30a,4e)-Lossless

a

E2,C2

14a

21a

35a

E3

a

17a

Is this really explicit?<E2,C2> : {0,1}20a x {0,1}a → {0,1}14a x {0,1}21a

- E2 size is a fixed constant.
- Can be shown to exist with a simpleProbabilistic argument, and then be foundby exhaustive search.
- Same deal with E3.
- E1 needs to be arbitrarily large
- “Regular” (non-lossless) constant degree expander.
- Explicit constructions are known!
- For permutation property – need consistently labeled expander

Construction Analysis

- Claim: The resulting conductor
- E : {0,1}n x {0,1}2a → {0,1}n-30aIs an (n-30a,4e)-lossless conductor.
- d = 2a
- Lets follow entropy flow from input (X1X2,R2R3) to the output Y1Y3.
- Let k = H∞(X1,X2)
- Want to show that we end up with k+2a min-entropy.

- <E1,C1> & <E1,C2> conserve entropy. Therefore:
- k + a = H∞(X1,X2,R2)
- = H∞(X1,Y2,Z2)
- = H∞(Y1,Z1,Z2)

E3: (15a,e) Lossless Conductor

- If we prove H∞(Y1) ≥ k – 14a
- H∞(Z1,Z2 | Y1) ≤ 15a

E3 will transfer a bits of entropy without losses.

n2

n1

x2(20a)

x1(n-20a)

r2(a)

E2, C2

y2(14a)

E1, C1

z2(21a)

z1(14a)

r3(a)

E3

y3(17a)

y1(n-20a)

Recall Lemma A:

Let (X1,X2) be a prob. on the vertices of the Zig-Zag product.Given e>0 and a:

It suffices to consider only 2 extreme cases:

H∞(X2 | X1 = x1) ≥ a

H∞(X2 | X1 = x1) ≤ a

(in our case a = 14a)

n2

n1

x2(20a)

x1(n-20a)

Case I:

H∞(Y2 | X1 = x1) = 14a,

and is a good seed for E1.

Since H∞(X1) ≥ k – 20a,

and E1: is an

(6a,e) extracting Conductor

E1 transfers 6a bits of entropy from the seed into Y1, so:

H∞(Y1) ≥ k – 14a

r2(a)

E2, C2

y2(14a)

E1, C1

z2(21a)

z1(14a)

r3(a)

E3

y3(17a)

y1(n-20a)

Case II:

H∞(X1,X2) = k

H∞(X1) ≥ k - 14a

E2 is an extractor, so:

H∞(Y2|X1 = x1) > H∞(X2|X1 = x1)

H∞(X1,Y2) ≥ H∞(X1,X2) = k

<E1,C1> is a permutation, so:

also H∞(Y1,Z1) ≥ k.

Again we get:

H∞(Y1) ≥ k – 14a

Entropy flow

n2

n1

x2(20a)

x1(n-20a)

r2(a)

E2, C2

y2(14a)

E1, C1

z2(21a)

z1(14a)

r3(a)

E3

y3(17a)

y1(n-20a)

Entropy flow

To complete the analysis:

We have shown that for any

H∞(Z1,Z2|Y1 = y1) ≤

≤ H∞(Y1,Z1,Z2) - H∞(Y1) ≤

≤ (k + a) – (k – 14a) = 15a

- Lossless conductor E3 transfers a bits of entropy from R3 to Y3 :

H∞(Y3|Y1 = y1) ≥

≥ H∞(Z1,Z2|Y1 = y1) + a

H∞(Y1,Y3) = k + 2a

n2

n1

x2(20a)

x1(n-20a)

r2(a)

E2, C2

y2(14a)

E1, C1

z2(21a)

z1(14a)

E3

r3(a)

y3(17a)

y1(n-20a)

Details Omitted

- Ignored the small error e in the outputs of the conductors. (assumed e=0)
- Saw explicit construction of constant degree lossless expanders by a specific (non-optimal) construction example.
- For fully formal proof need to show for general parameters.

Summary and Open Problems

- Our Result: Constant-Degree Lossless Expanders.
- Main tools: randomness conductors, zig-zag product
- Further Research:
- The undirected case (being lossless from both sides).
- Better expansion yet? D-O(1)

The End

Based on slides of Salil Vadhan

Let (X1,X2) be a prob. dist. on a finite product space.

Given e>0 and a:

There exists a dist. (Y1,Y2) such that:

- (X1,X2) and (Y1,Y2) are e-close.
- (Y1,Y2) is a convex ofeach with min-entropy ≥ H∞(X1,X2) – log(1/e)

By this lemma it suffices to consider only 2 extreme cases:

- H∞(X2 | X1 = x1) “small”
- H∞(X2 | X1 = x1) “large”

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