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Randomness Conductors and Constant-Degree Expansion beyond the Degree / 2 Barrier. Salil Vadhan - Harvard University Michael Capalbo - DIMACS Omer Reingold - Weizmann Avi Wigderson - IAS. Slides by Guy Fridland. Outline. Introduction & Overview Randomness conductors

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randomness conductors and constant degree expansion beyond the degree 2 barrier

Randomness Conductors andConstant-Degree Expansionbeyond the Degree / 2 Barrier

Salil Vadhan - Harvard University

Michael Capalbo - DIMACS

Omer Reingold - Weizmann

Avi Wigderson - IAS

Slides by Guy Fridland

outline
Outline
  • Introduction & Overview
  • Randomness conductors
  • The Original Zig-Zag product & its limitations
  • The New Zig-Zag product for conductors
  • Explicit Construction
the big picture
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…
expander graphs

N

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|

expander graphs1

N

N

|X| K

D

|(X)|  A |X|

Expander Graphs

Goals:

  • 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).
this work constant degree lossless expanders

N

D

This Work: Constant-Degree “Lossless” Expanders

N

When:

A=(1-)∙D

0< small constant

we call the expander:

Lossless

|X| K

|(X)|  (1-) D |X|

why bother with the degree 2 barrier
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].
outline1
Outline
  • Introduction & Overview
  • Randomness conductors
  • The Original Zig-Zag product & its limitations
  • The New Zig-Zag product for conductors
  • Explicit Construction
slide9

almost

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]

slide10

Min-Entropy

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

maximum are taken over

slide11

Min-Entropy

  • 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)
randomness conductors

n-bit input

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)
slide13

Def: Conductor

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
slide14

n-bit input

“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
slide15
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

slide16

Special Conductors:

  • 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.
slide17

n-bit input

d-bit seed

Conductor

Buffer

ExtractingConductor

LosslessConductor

m-bit output

b-bit seed

Buffer Conductor

slide18

Def: Permutation 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

lossless conductors vs expanders

K ≤ Kmax

N

x

n-bit input

D=2d

(kmax,ε)Lossless

Conductor

d-bit

seed

M

y

(1-)DK

m-bit output

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

slide20

Lets recall why we’re here

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.
outline2
Outline
  • Introduction & Overview
  • Randomness conductors
  • The Original Zig-Zag product & its limitations
  • The New Zig-Zag product for conductors
  • Explicit Construction
slide24
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)

slide25

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
zig zag analysis case i

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.
zig zag analysis case ii

First small step does nothing.

  • Step on big graph “scatters” among clouds (shifts entropy)
  • Second small step adds entropy.
Zig-Zag Analysis (Case II)
  • Case II: Conditional distributions within clouds uniform.
slide28

Inherent Entropy Loss

  • 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.
outline3
Outline
  • Introduction & Overview
  • Randomness conductors
  • The Original Zig-Zag product & its limitations
  • The New Zig-Zag product for conductors
  • Explicit Construction
zig zag for conductors
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 conductors1
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 conductors2
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 conductors3
Zig-Zag for Conductors

n2

n1

d2

E2

Finally:

E3: Deliver entropy lost in E1 & E2 to the output

d1

E1

d3

E3

n1

m3

zig zag for conductors4
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 conductors5
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
Analysis of Entropy Flow

Case I:

Conditional

entropy within

clouds large.

n2

n1

d2

E2,C2

d1

E1,C1

b2

b1

d3

E3

n1

m3

analysis of entropy flow1
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

outline4
Outline
  • Introduction & Overview
  • Randomness conductors
  • The Original Zig-Zag product & its limitations
  • The New Zig-Zag product for conductors
  • Explicit Construction
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 construction1
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

is this really explicit

20a

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
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.
slide43

Entropy flow

  • <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)

slide44
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)

slide45

Entropy flow

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)

slide46
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)

slide47

Entropy flow

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
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
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

slide50

Lemma A:

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”