Constant Degree, Lossless Constant Degree, Lossless ExpandersExpanders

Omer Reingold

AT&T

joint work with Michael Capalbo (IAS),

Salil Vadhan (Harvard),

and Avi Wigderson (Hebrew U., IAS)

Expander Graphs (Balanced Case)Expander Graphs (Balanced Case)

|(S)| >A |S|S, |S| K

An innocent looking object … but intimately related to various fundamental problems (Network Design, Complexity and Proof Theory, Derandomization, Coding Theory, Cryptography, ...)

D

N N

Expander Graphs (Balanced Case)Expander Graphs (Balanced Case)

|(S)| >A |S|S, |S| K

How large can A be? • Trivial upper bound: A D. • Random graphs: AD.• Previously, best explicit expanders: A =D/2

(for constant D and “large” K).

D

N N

This Work: Const. Degree, Lossless This Work: Const. Degree, Lossless Expanders …Expanders …

… that may even be slightly unbalanced:

|(S)| >(1-) D |S|D

N M= N

S, |S| K

0<, 1 are constants D is constant & K= (N)

For the very curious only:K= ( M/D) & D= poly log (1/( )) (fully explicit: D= quasi poly log(1/( ) )).

A Bit of ContextA Bit of Context

• Explicit construction of constant degree expanders is difficult.

• Celebrated sequence of algebraic constructions [Mar73 ,GG80,JM85,LPS86,AGM87,Mar88,Mor94].

• Ramanujan graphs with expansion D/2 [Kahale95].

• “Combinatorial” constructions: Ajtai [Ajt87], more explicit and very simple: [RVW00].

• “Lossless objects”: [Alo95,RR99,TUZ01*]• Unique neighbor, constant degree expanders [Cap01].

Why Bother with the Deg./2 Barrier?Why Bother with the Deg./2 Barrier?

• Because its there ???

• 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 [BMRRS00].

– Various storage schemes [UW88,BMRS00].

– Hard tautologies for various proof systems

[BW99,ABRW00,AR01].

Distributed routing in networksDistributed routing in networks

The Task [[PU89,ALM96,BFU99PU89,ALM96,BFU99]]: Finding vertex/edge disjoint paths in a distributed manner. Much easier if the network is composed of lossless expanders.

Distributed routing in networksDistributed routing in networks

Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10.

|S| K

OutputsInputs

...

Step 1: Match to “unique neighbors” of S

Then, continue with (at most |S|/10) unmatched vertices in S

...

Distributed routing in networksDistributed routing in networks

Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10.

|S| K

OutputsInputs

Incredibly Fault TolerantIncredibly Fault Tolerant [[UW87UW87]]: Works even if adversary removes 3/4 of D edges from each vertex.

...

Simple Expander Codes Simple Expander Codes [G63,Z71,ZP76,T81,SS96]

M= N (Parity Checks)

Linear code. Rate 1 - M/N = 1 -

Minimum distance K.

Relative distance K/N= ( / D) = / poly log (1/).

For small beats the Zyablov bound and is quite close to the Gilbert-Varshamov bound of log (1/).

N (Variables)

Fix =1/10 :

Sets of size K= ( N/D) expand by a factor 9D/10.

D

1

100

1

++

+

+

0

Error set B, |B| K/2

• Algorithm: At each phase, flip every variable that “sees” a majority of 1’s (i.e, unsatisfied constraints).

Simple Decoding Algorithm in Linear Simple Decoding Algorithm in Linear TimeTime (& log n parallel phases) [SS 96]

M= N (Constraints)N (Variables)

++

+

+1

100

1|Flip\B| |B|/4.|B\Flip| |B|/4. |Bnew| |B|/2.

|(B)| > (1-) D |B|

|(B)Sat| < 2 D |B|0

10

0

11

0

Hints Into the Expander ConstructionHints Into the Expander Construction

• Starting point [RVW00]: A simple combinatorial construction of constant-degree expanders with simple analysis.

• The heart of the construction – New Graph Product: 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.

The Zig-Zag Product [RVW00]

z

Thm. If G1 is a “good” expander, then Expansion (G1 G2) Expansion (G2)z

Zig-Zag Analysis (Case I)Zig-Zag Analysis (Case I) [RVW00]

In Case I, the second “small step” is guaranteed to expand. The first may be “lost”.

In Case II, the reversed picture Need both small steps.

Trying to improveTrying to improve

???

???

Zig-Zag for Unbalanced GraphsZig-Zag for Unbalanced Graphs

• Second eigenvalue analysis for expanders – probably not useful in the unbalanced case.

• Extractors [NZ93] and condensers (under various formalizations [RR99,RSW00,TUZ01]), work well in the unbalanced case.

• In fact, [RVW00] analyzed a zig-zag product for extractors (with an “easier goal”).

• We introduce randomness conductors that interpolate expanders, extractors, condensers & hash functions, and analyze the zig-zag product for conductors.

Randomness ConductorsRandomness Conductors

• Expanders, extractors, condensers & hash functions are all functions, f : [N] [D] [M], that transform:

S “of entropy” k S’ = f (S,Uniform) “of entropy” k’

• Many flavors:

– Measure of entropy.– Balanced vs. unbalanced.– Lossless vs. lossy.– Lower vs. upper bound on k.– Is S’ close to uniform?

– …

Randomness conductors:

As in extractors.

Allows the entire spectrum.

On the Board ?On the Board ?

• Randomness conductors -- a space of combinatorial objects:– From Expanders to Extractors in a few easy steps.– On measures of entropy. – The definition of randomness conductors.– Previous constructions and composition

techniques from the extractor literature extend to (useful) explicit constructions of conductors.

• The zig-zag product for conductors can produce constant degree, lossless expanders.

Summary and Open ProblemsSummary and Open Problems• Our Result: (Slightly Unbalanced), Constant Degree, (Slightly Unbalanced), Constant Degree,

Lossless ExpandersLossless Expanders.

• Seen: some applications, hints into the construction, and a short encounter with randomness conductors.

Further Research:

• The undirected case (being lossless from both sides).• Better expansion yet?• Continue the study of randomness condensers.

Definition: Randomness ConductorsDefinition: Randomness Conductors

• For any function : [0, log N] [0, log D] [0,1], the function f : [N] [D] [M], is an - conductor if: k, k’,

S, of min entropy k

f

Uniform

S’ = f (S,Uniform)

S’ is - close to min entropy” k’

(min entropy k x, Pr[x] 2-k)

Lossless Expanders are Incredibly Lossless Expanders are Incredibly Fault Tolerant Fault Tolerant [[UW87UW87]]

Let an adversary remove (1-) D edges for each vertex.• Still expands by a factor (1- / ) D’ !!

|(S)| >(1-) |S|S, |S| K

D

N N