Randomness Conductors

Expander Graphs

Randomness Extractors

Condensers

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

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’

PlanDefinitions & Applications:• The balanced case (M = N).

– Vertex Expansion.– 2nd Eigenvalue Expansion.

• The unbalanced case (M ≪ N).– Extractors, Dispersers, Condensers.

Conductors• Universal Hash Functions.

Constructions:• Zigzag Product & Loosless Expanders.

(Bipartite) Expander Graphs(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

Important: every (not too large) set expands.

D

N N

(Bipartite) Expander Graphs(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

•Main goal: minimize D (i.e. constant D)

•Degree 3 random graphs are expanders! [Pin73]

D

N N

(Bipartite) Expander Graphs(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

Also: maximize A.• Trivial upper bound: A D

– even A ≲ D-1• Random graphs: AD-1

D

N N

Applications of ExpandersThese “innocent” objects are intimately

related to various fundamental problems:

• Network design (fault tolerance), • Sorting networks, • Complexity and proof theory,• Derandomization, • Error correcting codes, • Cryptography, • Ramsey theory • And more ...

Non-blocking Network with On-line Path Selection

[ALM]N (Inputs) N (Outputs)

Depth O(log N), size O(N log N), bounded degree.

Allows connection between input nodes and output nodes using vertex disjoint paths.

Non-blocking Network with On-line Path Selection

[ALM]N (Inputs) N (Outputs)

Every request for connection (or disconnection) is satisfied in O(log N) bit steps:

• On line. Handles many requests in parallel.

The Network

“Lossless” Expander

N (Inputs) N (Inputs)

Slightly Unbalanced, “Lossless” Expanders

|(S)| 0.9 D |S|S, |S| K D

N M= N

0< 1 is an arbitrary constant D is constant & K= (M/D) = (N/D). [CRVW 02]: such expanders (with D = polylog(1/))

Property 1: A Very Strong Unique Neighbor Property

S, |S| K, |(S)| 0.9 D |S|

SNon Unique neighbor

S has 0.8 D |S| unique neighbors !

Unique neighbor of S

Using Unique Neighbors Using Unique Neighbors for Distributed Routingfor Distributed Routing

Task: match S to its neighbors (|S| K)

S

Step I: match S to its unique neighbors.

S`

Continue recursively with unmatched vertices S’.

Reminder: The Network

Adding new paths: think of vertices used by previous paths as faulty.

Property 2: Incredibly Incredibly Fault TolerantFault Tolerant

S, |S| K, |(S)| 0.9 D |S|

Remains a lossless expander even if adversary removes (0.7 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) = / polylog (1/). For small beats the Zyablov bound and is quite

close to the Gilbert-Varshamov bound of / log (1/).

N (Variables)

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 Simple Decoding Algorithm in Linear TimeLinear Time (& 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)| > .9 D |B|

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

0

10

0

11

0

Random Walk on Random Walk on Expanders [AKS 87]Expanders [AKS 87]

...x0

x1

x2 xi

xi converges to uniform fast (for arbitrary x0).

For a random x0: the sequence x0, x1, x2 . . . has interesting “random-like” properties.

Expanders Add EntropyExpanders Add Entropy

Prob. dist. X

•Definition we gave: |Support(X’)| A |Support(X)|

•Applications of the random walk rely on “less naïve” measures of entropy.

•Almost all explicit constructions directly give “2nd eigenvalue expansion”.

•Can be interpreted in terms of Renyi entropy.

D

NM

x x’

Induced dist. X’

22ndnd Eigenvalue Expansion Eigenvalue Expansion

•P=(Pi,j) - transition probabilities matrix:Pi,j= (# edges between i and j in G) / D

•Goal: If [0,1]n is a (non-uniform) distribution on vertices of G, then P is “closer to uniform.”

D

G - Undirected

D

N

Symmetric

N

N

22ndnd Eigenvalue Expansion Eigenvalue Expansion0 1 … N-1, eigenvalues of P.

0 =1, Corresponding eigenvector v0 =1N:

P(Uniform)=Uniform– Second eigenvalue (in absolute value):

=(G)=max{|1|,|N-1|}

– G connected and non-bipartite <1 is a good measure of the expansion of G

[Tan84, AM84, Alo86]. Qualitatively:

G is an expander (G) < β < 1

Randomness ConductorsRandomness Conductors

• Expanders, extractors, condensers & 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.