Post on 26-Mar-2015
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.