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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.
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Randomness Conductors Meta-Definition N MProb. 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”.
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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
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Vertex Expansion N N |Support(X’)||Support(X)|≤ K D ≥ A |Support(X)| (A > 1) Lossless expanders: A > (1-ε) D (for ε < ½)
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2nd Eigenvalue Expansion N N X D X’λ < β < 1, collision(X’) –1/N ≤ λ2 (collision(X) –1/N)
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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’)).
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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
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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? – …
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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’
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ConstructionsMost 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).
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[CRVW 02]: Const. Degree, Lossless Expanders … N N∀S, |S|≤ K |Γ(S)| ≥ (1-ε) D |S| D (K=Ω (N))
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… That Can Even Be Slightly Unbalanced N M=δ N ∀S, |S|≤ K |Γ(S)| ≥ (1-ε) D |S| D0<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)).
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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].
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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.
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The Zigzag Product z“Theorem”:Expansion (G1 z G2) ≈ min {Expansion (G1), Expansion (G2)}
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Zigzag Intuition (Case I)Conditional distributions within “clouds” far from uniform – The first “small step” adds entropy. – Next two steps can’t lose entropy.
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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.
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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!
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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.
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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
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Inherent Entropy Loss– In each case, only one of two small steps “works”– But paid for both in degree.
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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.
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Some Open Problems• Being lossless from both sides (the non-bipartite case).• Better expansion yet?• Further study of randomness conductors.
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