2. Randomness Conductors –
Motivation
• 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.
3. Randomness Conductors
Meta-Definition
N
M
Prob. dist. X Prob. dist. X’
D
x x’
An R-conductor if for every (k,k’) ∈ R,
X has ≥ k bits of “entropy” ⇒
X’ has ≥ k’ bits of “entropy”.
4. Measures 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 ε-close if
maxT | Pr[X∈T] - Pr[Y∈T] | = ½ ||X-Y||1 ≤ ε
• X’ is ε-close Y of min-entropy k ⇒ |
Support(X’)|≥ (1-ε) 2k
5. Vertex Expansion
N N
|Support(X’)|
|Support(X)|≤ K D ≥ A |Support(X)|
(A > 1)
Lossless expanders: A > (1-ε) D (for ε < ½)
7. Unbalanced Expanders /
Condensers
N M≪N
X X’
D
• 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’)).
8. Dispersers and Extractors
[Sipser 88,NZ 93]
N M≪N
X X’
D
• (k,ε)-disperser if
|Support(X)| ≥ 2k ⇒ |Support(X’)|≥ (1-ε) M
• (k,ε)-extractor if
Min-entropy(X) ≥ k ⇒ X’ ε-close to uniform
9. Randomness 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’
Randomness conductors:
• Many flavors:
– Measure of entropy. As in extractors.
– Balanced vs. unbalanced.
– Lossless vs. lossy.
Allows the entire
– Lower vs. upper bound on k.
spectrum.
– Is X’ close to uniform?
– …
10. Conductors: Broad Spectrum
Approach
N M≪N
X X’
D
• An ε-conductor, ε:[0, log N]×[0, log M]→[0,1],
if: ∀ k, k’, min-entropy(X’) ≥ k ⇒
X’ ε (k,k’)-close to some Y of min-entropy k’
11. 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).
12. [CRVW 02]: Const. Degree,
Lossless Expanders …
N N
∀S, |S|≤ K |Γ(S)| ≥ (1-ε) D |S|
D
(K=Ω (N))
13. … That Can Even Be Slightly
Unbalanced
N M=δ N
∀S, |S|≤ K |Γ(S)| ≥ (1-ε) D |S|
D
0<ε,δ≤ 1 are constants ⇒ D is constant & K=Ω (N)
For the curious:
K=Ω (ε M/D) & D= poly (1/ε, log (1/δ)) (fully
explicit: D= quasi poly (1/ε, log (1/δ)).
14. History
• 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].
15. The 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.
16. The Zigzag Product
z
“Theorem”:
Expansion (G1 z G2) ≈ min {Expansion (G1), Expansion (G2)}
17. Zigzag Intuition (Case I)
Conditional distributions within “clouds” far from uniform
– The first “small step” adds entropy.
– Next two steps can’t lose entropy.
18. 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.
19. Reducing to the Two Cases
• Need to show: the transition prob. matrix M
of G1 z 2 shrinks every vector π∈ℜND that is
G
perp. to uniform.
1 2 … … D
• Write π as N×D Matrix: 1
π ⊥ uniform ⇒ sum of …
entries is 0. u .4 -.3 … … 0
– RowSums(x) = “distribution” …
on clouds themselves N
• Can decompose π = π|| + π⊥ , where π|| is constant on rows,
and all rows of π⊥ are perp. to uniform.
• Suffices to show M shrinks π|| and π⊥ individually!
20. 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.
21. Closer Look: Rotation Maps
• Expanders normally viewed as maps
(vertex)×(edge label) → (vertex).
X,i
Y,j • Here: (vertex)×(edge label) →
(vertex)×(edge label).
Permutation ⇒ The big step never lose.
(X,i) → (Y,j) if
(X, i ) and (Y, j ) Inspired by ideas from the setting of
correspond to “extractors” [RR99].
same edge of G1
22. Inherent Entropy Loss
– In each case, only one of two small steps “works”
– But paid for both in degree.
24. Zigzag for Unbalanced
Graphs
• 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.
25. Some Open Problems
• Being lossless from both sides (the
non-bipartite case).
• Better expansion yet?
• Further study of randomness
conductors.
