Randomness Conductors (II)

1. Condensers Expander Graphs Universal Hash Functions . . . . . . . . . . . . Randomness Extractors Randomness Conductors (II)

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. N x’ Randomness Conductors Meta-Definition An R-conductor if for every (k,k’)  R, X has k bits of “entropy” X’ has k’ bits of “entropy”. M Prob. dist. X Prob. dist. X’ D x

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 -close if maxT | Pr[XT] - Pr[YT] | = ½ ||X-Y||1   • X’ is -close Y of min-entropyk|Support(X’)| (1-) 2k

5. N N |Support(X’)| A |Support(X)|(A > 1) |Support(X)| K D Vertex Expansion Lossless expanders: A > (1-) D (for  < ½)

6. N N D 2nd Eigenvalue Expansion  < β < 1,collision(X’) –1/N 2 (collision(X) –1/N) X X’

7. N D Unbalanced Expanders / Condensers M ≪N X X’ • 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. N D Dispersers and Extractors [Sipser 88,NZ 93] M ≪N X X’ • (k,)-disperser if |Support(X)|  2k|Support(X’)| (1-) M • (k,)-extractor if Min-entropy(X)  kX’ -close to uniform

9. Randomness conductors: • As in extractors. • Allows the entire spectrum. 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’ • Many flavors: • Measure of entropy. • Balanced vs. unbalanced. • Lossless vs. lossy. • Lower vs. upper bound on k. • Is X’ close to uniform? • …

10. N D Conductors: Broad Spectrum Approach M ≪N X X’ • 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-entropyk’

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. N N D [CRVW 02]: Const. Degree, Lossless Expanders … S, |S| K (K= (N)) |(S)| (1-) D |S|

13. N D … That Can Even Be Slightly Unbalanced M= N S, |S| K |(S)| (1-) D |S| 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/)).

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. z z • “Theorem”: • Expansion (G1 G2)  min {Expansion (G1), Expansion (G2)} The Zigzag Product

17. The first “small step” adds entropy. Zigzag Intuition (Case I) Conditional distributions within “clouds” far from uniform • Next two steps can’t lose entropy.

18. First small step does nothing. • Step on big graph “scatters” among clouds (shifts entropy) • Second small step adds entropy. Zigzag Intuition (Case II)Conditional distributions within clouds uniform

19. z Reducing to the Two Cases • Need to show: the transition prob. matrix M ofG1 G2 shrinks every vector ND that is perp. to uniform. • Write  as ND 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!

20. Results & Extensions [RVW00] • Simple analysis in terms of second eigenvalue mimics the intuition. • Can obtain degree3 ! • 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. 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 Closer Look: Rotation Maps • (X,i) (Y,j) if • (X, i ) and (Y, j ) correspond to same edge of G1

22. Inherent Entropy Loss • In each case, only one of two small steps “works” • But paid for both in degree.

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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.