Randomness Conductors (II)

Expander Graphs

Randomness Extractors

Condensers

Universal Hash Functions............

Randomness Conductors – Randomness Conductors – MotivationMotivation

• Various relations between expanders, extractors, condensers & universal hash functions.

• Unifying all of these as instances of a more general combinatorial object:– Useful in constructions.– Possible to study new phenomena not

captured by either individual object.

Randomness Conductors Randomness Conductors Meta-DefinitionMeta-Definition

Prob. dist. X

An R-conductor if for every (k,k’) R, X has k bits of “entropy” X’ has k’ bits of “entropy”.

D

NM

x x’

Prob. dist. X’

Measures of EntropyMeasures of Entropy•A naïve measure - support size

•Collision(X) = Pr[X(1)=X(2)] = ||X||2

•Min-entropy(X) k if x, Pr[x] 2-k

•X and Y are -close if maxT | Pr[XT] - Pr[YT] | = ½ ||X-Y||1

•X’ is -close Y of min-entropy k |Support(X’)| (1-) 2k

Vertex ExpansionVertex Expansion

|Support(X’)| A |Support(X)|

(A > 1)

|Support(X)| K D

N N

Lossless expanders: A > (1-) D (for < ½)

22ndnd Eigenvalue Expansion Eigenvalue Expansion

X’X D

N N

< β < 1, collision(X’) –1/N 2 (collision(X) –1/N)

Unbalanced Expanders / Condensers

X’X D

N M ≪ N

• Farewell constant degree (for any non-trivial task |Support(X)|= N0.99, |Support(X’)| 10D)

• Requiring small collision(X’) too strong (same for large min-entropy(X’)).

Dispersers and Extractors [Sipser 88,NZ 93]

X’X D

N M ≪ N

• (k,)-disperser if |Support(X)| 2k |Support(X’)| (1-) M

• (k,)-extractor if Min-entropy(X) k X’ -close to uniform

Randomness ConductorsRandomness Conductors

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

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

• Many flavors:– Measure of entropy.– Balanced vs. unbalanced.– Lossless vs. lossy.– Lower vs. upper bound on k.– Is X’ close to uniform?– …

Randomness conductors:

As in extractors.

Allows the entire spectrum.

Conductors: Broad Spectrum Approach

X’X D

N M ≪ N

• An -conductor, :[0, log N][0, log M][0,1], if: k, k’, min-entropy(X’) k X’ (k,k’)-close to some Y of min-entropy k’

Constructions

Most applications need explicit expanders. Could mean:• Should be easy to build G (in time poly N).• When N is huge (e.g. 260) need:

– Given vertex name x and edge label i easy to find the ith neighbor of x (in time poly log N).

[CRVW 02]: Const. Degree, Lossless Expanders …

|(S)| (1-) D |S|S, |S| K (K= (N))

D

N N

… That Can Even Be Slightly Unbalanced

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

N M= N

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

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

HistoryHistory• Explicit construction of constant-degree expanders

was difficult.

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

• Achieved optimal 2nd eigenvalue (Ramanujan graphs), but this only implies expansion D/2 [Kah95].

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

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

[Cap01,AC02].

The Lossless ExpandersThe Lossless Expanders

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

• Heart of construction – New Zig-Zag 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 Zigzag Product

z

“Theorem”:

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

z

Zigzag Intuition (Case I) Zigzag Intuition (Case I) Conditional distributions within “clouds” far from uniform

– The first “small step” adds entropy.

– Next two steps can’t lose entropy.

Zigzag Intuition (Case II)Zigzag Intuition (Case II) Conditional distributions within clouds uniform

• First small step does nothing.

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

• Second small step adds entropy.

Reducing to the Two Cases

• Need to show: the transition prob. matrix M of G1 G2 shrinks every vector ND that is perp. to uniform.

• Write as ND Matrix: uniform sum of

entries is 0.– RowSums(x) =

“distribution” on clouds themselves

• Can decompose = || + , where || is constant on rows, and all rows of are perp. to uniform.

• Suffices to show M shrinks || and individually!

z

1 2 … … D

1

…

u .4 -.3 … … 0

…

N

Results & Extensions [RVW00]

• Simple analysis in terms of second eigenvalue mimics the intuition.

• Can obtain degree 3 !• Additional results (high min-entropy

extractors and their applications).

• Subsequent work [ALW01,MW01] relates to semidirect product of groups new results on expanding Cayley graphs.

Closer Look: Rotation Maps

• Expanders normally viewed as maps (vertex)(edge label) (vertex).

• Here: (vertex)(edge label) (vertex)(edge label).

Permutation The big step never lose.

Inspired by ideas from the setting of “extractors” [RR99].

X,i

Y,j

(X,i) (Y,j) if(X, i ) and (Y, j ) correspond to same edge of G1

Inherent Entropy LossInherent Entropy Loss

– In each case, only one of two small steps “works”

– But paid for both in degree.

Trying to improveTrying to improve

???

???

Zigzag for Unbalanced Zigzag for Unbalanced GraphsGraphs

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

• Previous constructions and composition techniques from the extractor literature extend to (useful) explicit constructions of conductors.

Some Open ProblemsSome Open Problems• Being lossless from both sides (the

non-bipartite case).• Better expansion yet?• Further study of randomness

conductors.