randomness conductors (ii) expander graphs randomness extractors condensers universal hash functions
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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.
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