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Randomness Conductors and Constant-Degree Expansion beyond the Degree / 2 Barrier

Randomness Conductors and Constant-Degree Expansion beyond the Degree / 2 Barrier. Salil Vadhan - Harvard University Michael Capalbo - DIMACS Omer Reingold - Weizmann Avi Wigderson - IAS. Slides by Guy Fridland. Outline. Introduction & Overview Randomness conductors

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Randomness Conductors and Constant-Degree Expansion beyond the Degree / 2 Barrier

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  1. Randomness Conductors andConstant-Degree Expansionbeyond the Degree / 2 Barrier Salil Vadhan - Harvard University Michael Capalbo - DIMACS Omer Reingold - Weizmann Avi Wigderson - IAS Slides by Guy Fridland

  2. Outline • Introduction & Overview • Randomness conductors • The Original Zig-Zag product & its limitations • The New Zig-Zag product for conductors • Explicit Construction

  3. The big picture Expanders. What are they good for? • High Connectivity. • No “bottle-neck” Many applications! • Network design. • Sorting. • Complexity theory • Cryptography. • Coding theory • Proof complexity • Many more…

  4. N D Expander Graphs • Bipartite graph G with: • N inputs, N outputs • Every input connected to D • outputs. • Saw many definitions: • Algebric (spectral gap) • (1+d)-expander • Definition : (K,A)-expander • The graph is (K,A)-expander if every set X of at most K • inputs is connected to at least A∙|X| outputs. N |X| K |(X)|  A |X|

  5. N N |X| K D |(X)|  A |X| Expander Graphs Goals: • Minimize the degree D • Maximize the expansion A Q: How large can A be? • Trivial upper bound: A  D. • Random graphs: A=D-1.01 • But many applications need explicit (deterministic & efficient) constructions. • Previously, best explicit expanders: A =D/2 • Algebric construction: optimal 2nd eigenvalue (Ramanujan graphs).

  6. N D This Work: Constant-Degree “Lossless” Expanders N When: A=(1-)∙D 0< small constant we call the expander: Lossless |X| K |(X)|  (1-) D |X|

  7. Why Bother with theDegree/2 Barrier? • For most applications of expanders: the more expansion the better. • Specific applications for lossless expanders: • Distributed routing in networks [PU89,ALM96,BFU99]. • Expander codes [Gal63,Tan81,SS96,Spi96,LMSS01]. • “Bitprobe complexity” of storing subsets [BMRS00]. • Distributed storage schemes [UW87]. • Hard tautologies for various proof systems [BW99,ABRW00,AR01].

  8. Outline • Introduction & Overview • Randomness conductors • The Original Zig-Zag product & its limitations • The New Zig-Zag product for conductors • Explicit Construction

  9. almost Measures of Expansion • Set Expansion: No small cuts, high connectivity • Expansion Factor: • Algebraic: Small second eigenvalue • Thm [...]: All equivalent. Each measure has its limitations. Extremely useful to adopt stronger measure of “randomness” Min-Entropy ! [Zuckerman `90]

  10. Min-Entropy • Min-EntropyMeasures how much a dist. is close to uniform. • Reminder: • Def: Min-Entropy of a dist. X: maximum are taken over

  11. Min-Entropy • Def: X is a k-source if H∞(X) ≥ k. • i.e. the uniform dist. on a set of size 2k is a k-source. • Def: Xis a (k,ε)-sourceif it ise-close to somek-source. • From now on: • N=2n, M=2m ,D=2d… • Def: X, Ye-close if D(X,Y)≤ ε • Measure of closeness: statistical difference: (D)

  12. n-bit input CON d-bit seed m-bit output Randomness Conductors • General framework for“randomness enhancing” functions: • Expanders, extractors,condensers, and theirrelatives… All of the above may be viewed as a function: E : [N] x [D]→[M] • Given guarantees on the randomness of the input dist. X. •  E gives guarantees on the randomness of the dist. E(X,UD)

  13. Def: Conductor A function Is a (kmax ,a,ε)-conductor, if for any 0≤k≤kmax, and any k-source X over {0,1}n,the dist. E(X,Ud) is a (k+a,ε)-source. E : {0,1}n x {0,1}d → {0,1}m n-bit input “k amount of entropy” CON d-seed m-bit output e-close to dist. with ≥ k+a amount of entropy Simple Conductor • E gets 2 inputs: • X: dist. With min-entropy k≤kmax • The uniform dist. UD (min-entropy d) • Output: • e-close to dist. with at leastk+a amount of entropy

  14. n-bit input “m-a amount of entropy” ExtCon d-seed m-bit output e-close to dist. with ≥ m amount of entropy Special Conductors: Def: Extracting Conductor A function Is a(a,ε)-extracting conductor if it is a (m-a,a,ε)-conductor. E : {0,1}n x {0,1}d → {0,1}m • Kmax = m-a • If input entropy is m-a •  output e-close to uniform

  15. a = d input entropy is k Seed entropy is d  output e-close to dist. With≥ k+d amout of entropy. n-bit input “k amount of entropy” LossLessCon d-seed m-bit output e-close to dist. with ≥ k+damount of entropy Special Conductors: Def: Lossless Conductor A function Is a(kmax,ε)-Lossless conductor if it is a (kmax,a,ε)-conductor. E : {0,1}n x {0,1}d → {0,1}m

  16. Special Conductors: • Next 2 conductors combine the above two cases(extracting & lossless) Def: Buffer Conductor A function Is a(a,ε)-buffer conductor if E is a (a,ε)-extracting conductor & is an (kmax,ε)-lossless conductor. <E,C> : {0,1}n x {0,1}d → {0,1}m x {0,1}b • Intuition: • Think of a “Buffer conductor” as putting a bucket beneath a lossy conductor. • So when we pour randomness (water) into it, the leftovers (unused randomness), are stored for later use.

  17. n-bit input d-bit seed Conductor Buffer ExtractingConductor LosslessConductor m-bit output b-bit seed Buffer Conductor

  18. Def: Permutation Conductor The same as Buffer Conductor where n+d=m+b, & is a permutaion over {0,1}n+d n-bit input d-bit input Conductor • Is a permutation Buffer •  is alossless conductor(lossless even if the seed is ‘bad’ because it is apermutation) m-bit input b-bit input

  19. K ≤ Kmax N x n-bit input D=2d (kmax,ε)Lossless Conductor d-bit seed M y (1-)DK m-bit output Lossless Conductors Vs Expanders • x{0,1}ny{0,1}mif LossCond(x,r)=y for some r {0,1}d • (kmax,ε)-Lossless Conductor (2kmax,ε)-Lossless Expander 

  20. Lets recall why we’re here Explicit construction of a constant degreeLossless expander! • Starting Point: Zig-Zag Graph Product [RVW00] 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. • Composition of Expanders via the Zig-Zag Product will imply constant degree.

  21. Outline • Introduction & Overview • Randomness conductors • The Original Zig-Zag product & its limitations • The New Zig-Zag product for conductors • Explicit Construction

  22. z The Zig-Zag Product [RVW00]

  23. The Zig-Zag Product [RVW00]

  24. Size: n*d1 Degree: d22 A step on Gi “adds ai bits of entropy”.  A step on G “adds min{a1,a2} bits of entropy Suboptimal expansion! Deficiency Can be traced back to the expander composition itself. z • G = G1 G2 • Expansion ≈ min{Expansion(G1), Expansion(G2)} The Zig-Zag Product - Analysis G1: (n,d1) G2: (d1,d2)

  25. The Zig-Zag Product - Analysis Lemma A: Let (X1,X2) be a prob. on the vertices of the Zig-Zag product. It suffices to consider only 2 extreme cases: • H∞(X2 | X1 = x1) close to uniform • H∞(X2 | X1 = x1) far from uniform

  26. First step on small graph adds entropy. Zig-Zag Analysis (Case I) • Case I: Conditional distributions within “clouds” far fr. uniform. • Next two steps can’t lose entropy.

  27. First small step does nothing. • Step on big graph “scatters” among clouds (shifts entropy) • Second small step adds entropy. Zig-Zag Analysis (Case II) • Case II: Conditional distributions within clouds uniform.

  28. Inherent Entropy Loss • In each case, only one of two small steps adds entropy • But paid for both in degree. • Expansion is too low for lossless expanders • The new Zig-Zag product for conductors manages to avoid this problem.

  29. Outline • Introduction & Overview • Randomness conductors • The Original Zig-Zag product & its limitations • The New Zig-Zag product for conductors • Explicit Construction

  30. Zig-Zag for Conductors n1 • Start with a lossy conductor E1 • Any constant degree expander will do. • Goal: make it lossless. d1 E1 n1

  31. Zig-Zag for Conductors n2 n1 • “Small” Conductor E2: • Blow up each vertexto cloud of size 2n2 • Apply E2 to select edge for E1 • If input entropy large enough, output close to uniform • If E2 input entropy too large,we lose entropy! d2 E2 d1 E1 n1

  32. Zig-Zag for Conductors n2 n1 d2 E2 d1 E1 Idea: Keep “buffers” to retain lost entropy In lossy Conductors E1 & E2 n1

  33. Zig-Zag for Conductors n2 n1 d2 E2 Finally: E3: Deliver entropy lost in E1 & E2 to the output d1 E1 d3 E3 n1 m3

  34. Zig-Zag for Conductors • Let’s get formal:To define the product we need 3 objects: • <E1,C1> : {0,1}n1 x {0,1}d1 → {0,1}m1 x {0,1}b1Permutation Conductor • <E2,C2> : {0,1}n2 x {0,1}d2 → {0,1}d1 x {0,1}b2 Buffer Conductor • E3 : {0,1}b1+b2 x {0,1}d3 → {0,1}m3 Lossless Conductor • The Zig-Zag product for Conductors produces: E : {0,1}n x {0,1}d → {0,1}m n = n1+n2d = d2+d3m = m1+m3

  35. Zig-Zag for Conductors n2 n1 <E1,C1>: Permutation Conductor d2 E2,C2 <E2,C2>: Buffer Conductor d1 E1,C1 <E3,C3>: Lossless Conductor b2 b1 d3 E3 m1 m3

  36. Analysis of Entropy Flow Case I: Conditional entropy within clouds large. n2 n1 d2 E2,C2 d1 E1,C1 b2 b1 d3 E3 n1 m3

  37. Analysis of Entropy Flow Case II: Conditional entropy within clouds small. n2 n1 d2 E2,C2 d1 E1,C1 b2 b1 d3 E3 n1 m3

  38. Outline • Introduction & Overview • Randomness conductors • The Original Zig-Zag product & its limitations • The New Zig-Zag product for conductors • Explicit Construction

  39. Explicit Construction • Fix : a = 1000log(1/e) d = 2a; • (n-30a,6a,e)-Permutation Conductor • <E1,C1> : {0,1}n-20a x {0,1}14a → {0,1}n-20 x {0,1}14a • (14a,0,e)- Buffer Conductor • <E2,C2> : {0,1}20a x {0,1}a → {0,1}14a x {0,1}21a • (15a,e)-Lossless Conductor • E1 : {0,1}35a x {0,1}a → {0,1}17a • Claim: The resulting conductor • E : {0,1}n x {0,1}2a → {0,1}n-3aIs an (n-30a,4e)-lossless conductor.

  40. Explicit Construction n2 n1 E1: (n-30a,6a,e)-Permutation x2(20a) x1(n-20a) r2(a) E2: (14a,0,e)- Buffer E2,C2 E3: (15a,e)-Lossless y2(14a) E1,C1 C1: keep the whole seed (14a) z2(21a) z1(14a) C2: keep the (small) input & seed(20a +a = 21a) E3 y3(17a) y1(n-20a) E: (n-30a,4e)-Lossless

  41. 20a a E2,C2 14a 21a 35a E3 a 17a Is this really explicit? <E2,C2> : {0,1}20a x {0,1}a → {0,1}14a x {0,1}21a • E2 size is a fixed constant. • Can be shown to exist with a simpleProbabilistic argument, and then be foundby exhaustive search. • Same deal with E3. • E1 needs to be arbitrarily large • “Regular” (non-lossless) constant degree expander. • Explicit constructions are known! • For permutation property – need consistently labeled expander

  42. Construction Analysis • Claim: The resulting conductor • E : {0,1}n x {0,1}2a → {0,1}n-30aIs an (n-30a,4e)-lossless conductor. • d = 2a • Lets follow entropy flow from input (X1X2,R2R3) to the output Y1Y3. • Let k = H∞(X1,X2) • Want to show that we end up with k+2a min-entropy.

  43. Entropy flow • <E1,C1> & <E1,C2> conserve entropy. Therefore: • k + a = H∞(X1,X2,R2) • = H∞(X1,Y2,Z2) • = H∞(Y1,Z1,Z2) E3: (15a,e) Lossless Conductor • If we prove H∞(Y1) ≥ k – 14a • H∞(Z1,Z2 | Y1) ≤ 15a E3 will transfer a bits of entropy without losses. n2 n1 x2(20a) x1(n-20a) r2(a) E2, C2 y2(14a) E1, C1 z2(21a) z1(14a) r3(a) E3 y3(17a) y1(n-20a)

  44. Recall Lemma A: Let (X1,X2) be a prob. on the vertices of the Zig-Zag product.Given e>0 and a: It suffices to consider only 2 extreme cases: H∞(X2 | X1 = x1) ≥ a H∞(X2 | X1 = x1) ≤ a (in our case a = 14a)

  45. Entropy flow n2 n1 x2(20a) x1(n-20a) Case I: H∞(Y2 | X1 = x1) = 14a, and is a good seed for E1. Since H∞(X1) ≥ k – 20a, and E1: is an (6a,e) extracting Conductor  E1 transfers 6a bits of entropy from the seed into Y1, so: H∞(Y1) ≥ k – 14a r2(a) E2, C2 y2(14a) E1, C1 z2(21a) z1(14a) r3(a) E3 y3(17a) y1(n-20a)

  46. Case II: H∞(X1,X2) = k  H∞(X1) ≥ k - 14a E2 is an extractor, so: H∞(Y2|X1 = x1) > H∞(X2|X1 = x1)  H∞(X1,Y2) ≥ H∞(X1,X2) = k <E1,C1> is a permutation, so: also H∞(Y1,Z1) ≥ k. Again we get: H∞(Y1) ≥ k – 14a Entropy flow n2 n1 x2(20a) x1(n-20a) r2(a) E2, C2 y2(14a) E1, C1 z2(21a) z1(14a) r3(a) E3 y3(17a) y1(n-20a)

  47. Entropy flow Entropy flow To complete the analysis: We have shown that for any H∞(Z1,Z2|Y1 = y1) ≤ ≤ H∞(Y1,Z1,Z2) - H∞(Y1) ≤ ≤ (k + a) – (k – 14a) = 15a • Lossless conductor E3 transfers a bits of entropy from R3 to Y3 : H∞(Y3|Y1 = y1) ≥ ≥ H∞(Z1,Z2|Y1 = y1) + a  H∞(Y1,Y3) = k + 2a  n2 n1 x2(20a) x1(n-20a) r2(a) E2, C2 y2(14a) E1, C1 z2(21a) z1(14a) E3 r3(a) y3(17a) y1(n-20a)

  48. Details Omitted • Ignored the small error e in the outputs of the conductors. (assumed e=0) • Saw explicit construction of constant degree lossless expanders by a specific (non-optimal) construction example. • For fully formal proof need to show for general parameters.

  49. Summary and Open Problems • Our Result: Constant-Degree Lossless Expanders. • Main tools: randomness conductors, zig-zag product • Further Research: • The undirected case (being lossless from both sides). • Better expansion yet? D-O(1) The End Based on slides of Salil Vadhan

  50. Lemma A: Let (X1,X2) be a prob. dist. on a finite product space. Given e>0 and a: There exists a dist. (Y1,Y2) such that: • (X1,X2) and (Y1,Y2) are e-close. • (Y1,Y2) is a convex ofeach with min-entropy ≥ H∞(X1,X2) – log(1/e) By this lemma it suffices to consider only 2 extreme cases: • H∞(X2 | X1 = x1) “small” • H∞(X2 | X1 = x1) “large”

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