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

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.

Randomness Conductors Meta-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 N M x x Prob. dist. X

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 max T | Pr[X T] - Pr[Y T] | = ½ ||X-Y|| 1 X is -close Y of min-entropy k |Support(X)| (1- ) 2 k

Vertex Expansion |Support(X)| A |Support(X)| (A > 1) |Support(X)| K D NN Lossless expanders: A > (1- ) D (for < ½)

2 nd Eigenvalue Expansion XX D NN < 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)|= N 0.99, |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)| 2 k |Support(X)| (1- ) M (k, )-extractor if Min-entropy(X) k X -close to uniform

Randomness Conductors Expanders, extractors, condensers & universal hash functions are all functions, f : [N] [D] [M], that transform: X of entropy k X = f (X,U niform ) 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 ) need: –Given vertex name x and edge label i easy to find the i th neighbor of x (in time poly log N).

[CRVW 02] : Const. Degree, Lossless Expanders … | (S)| (1- ) D |S| S, |S| K (K= (N)) D NN

… 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/ )).

History Explicit construction of constant-degree expanders was difficult. Celebrated sequence of algebraic constructions [Mar73,GG80,JM85,LPS86,AGM87,Mar88,Mor94]. Achieved optimal 2 nd 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 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 (G 1 G 2 ) min {Expansion (G 1 ), Expansion (G 2 )} 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 cant 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 G 1 G 2 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! z 12……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 G 1

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.

Some Open Problems Being lossless from both sides (the non-bipartite case). Better expansion yet? Further study of randomness conductors.