1 / 37

Expander Graphs, Randomness Extractors and List-Decodable Codes

Expander Graphs, Randomness Extractors and List-Decodable Codes. Salil Vadhan Harvard University Joint work with Venkat Guruswami (UW) & Chris Umans (Caltech). Connections in Pseudorandomness. List-Decodable Error-Correcting Codes. Pseudorandom Generators. [PV05,GR06]. This Work.

amberlyj
Download Presentation

Expander Graphs, Randomness Extractors and List-Decodable Codes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Expander Graphs,Randomness Extractorsand List-Decodable Codes Salil VadhanHarvard University Joint work with Venkat Guruswami (UW) & Chris Umans (Caltech)

  2. Connections in Pseudorandomness List-DecodableError-Correcting Codes PseudorandomGenerators [PV05,GR06] This Work [Tre99,TZ01,TZS01,SU01] [Tre99,RRV99,ISW99,SU01,U02] RandomnessExtractors This Work [GW94,WZ95,TUZ01,RVW00,CRVW02] [CW89,Z96] Expander Graphs Samplers

  3. Outline • Expander Construction • Application to Extractors • Connections • Conclusions

  4. S, |S| K D (Bipartite) Expander Graphs N Goals: • Minimize D • Maximize A • Minimize M M “(K,A) expander” |(S)|  A¢|S| Nonconstructive: • D = O(log(N/M)/) • A = (1-)¢D • M = O(KD/) O(1) if M=N = (log N) if M·pN

  5. Applications of Expanders • Fault-tolerant networks (e.g., [Pin73,Chu78,GG81]) • Sorting in parallel [AKS83] • Complexity theory [Val77,Urq87] • Derandomization [AKS87,INW94,Rei05,…] • Randomness extractors [CW89,GW94,TUZ01,RVW00] • Ramsey theory [Alo86] • Error-correcting codes [Gal63,Tan81,SS94,Spi95,LMSS01] • Distributed routing in networks [PU89,ALM96,BFU99]. • Data structures [BMRS00]. • Distributed storage schemes [UW87]. • Hard tautologies in proof complexity [BW99,ABRW00,AR01]. • Other areas of Math [KR83,Lub94,Gro00,LP01] Need explicit constructions (deterministic, time poly(log N)).

  6. S, |S| K Advantage of Expansion (1-)¢D • At least (1-2) D |S| elements of (S) areunique neighbors: touch exactly one edge from S N M |(S)| (1-) D |S| x D • Fault tolerance: Even if an adversary removes most (say ¾) edges from each vertex, lossless expansion maintained (with =4)

  7. D Application to Data Structures [BMRS00] N M • Goal: store small S½[N] s.t. can test membership by (probabilistically) reading 1 bit. • Expansion (1-)¢D )9 0,1 assignment to [M] s.t. for every x2[N], a 1-O() fraction of neighbors have correct answer! 0 0 0 1 S, |S| K /2 1 |(S)|  (1-)¢D¢|S| 0 1 1 0

  8. D Application to Data Structures [BMRS00] N M Size: M=O(K¢ log N) with optimal expander • (K¢log N) necessary to represent set. • Perfect hashing: same size, but read O(log N)-bit word 0 0 0 1 S, |S| K /2 1 0 1 1 0

  9. Explicit Constructions |right-side| M degree D expansion A >0 arbitrary constant. quasipoly(t)=exp(polylog t)

  10. S, |S| K D Our Result N M Thm: For every N, K, >0, 9explicit (K,A)expander with • degree D = poly(log N, 1/) • expansion A = (1-)¢D • #right vertices M = D2¢K1.01. |(S)|  A¢|S|

  11. Our Construction Left vertices = Fqn= polys of degree ·n-1 over Fq Degree = q Right vertices = Fqm+1 (f,y) =y’th neighbor of f =(y, f(y), (fh mod E)(y), (fh2 mod E)(y), …, (fhm-1 mod E)(y)) where E(Y) = irreducible poly of degree nh = a parameter Thm: This is a (K,A) expander with K=hm, A = q-hnm.

  12. Setting Parameters (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n N =Fqn , D =q, M =Fqm+1 Thm: This is a (K,A) expander with K=hm, A = q-hnm. Set h = poly(nm/), q = h1.01. Then: • D = q = poly(log N, 1/) • A = q-hnm ¸ (1-)¢ D • M = qm+1 = q¢ (h1.01)m = D¢ K1.01

  13. Rel’n to Parvaresh-Vardy Codes [PV05] (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm: This is a (K,A) expander with K=hm, A = q-hnm. • (f,y) = (y, y’th symbol of PV encoding f) • Proof of expansion inspired by list-decoding algorithm for PV codes.

  14. D List-Decoding View of Expanders N • For Tµ [M], define LIST(T) = {x2 [N] : (x)µT} • Lemma: G is a (=K,A) expander iff for all Tµ [M] of size AK-1, we have |LIST(T)| · K-1 M “(=K,A) expander” S, |S|=K |(S)|  A¢ K

  15. Comparing List-Decoding Views  : [N] £ [D]! [D]£[M] T µ [D]£[M]

  16. Proof of Expansion (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm:For A=q-nmh and any K·hm, we have TµFqm+1 of size AK-1)|LIST(T)|· K-1 Proof Outline (following [S97,GS99,PV05]): • Find a low-degree poly Q vanishing on T. • Show that every f 2LIST(T) is a “root” of a related polynomial Q’. • Show that deg(Q’) · K-1 =

  17. Proof of Expansion: Step 1 (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm: ForA=q-nmh, K=hm, |T|·AK-1)|LIST(T)|· K-1. Step 1: Find a low-degree poly Q vanishing on T. • Take Q(Y,Z1,…,Zm) to be of degree ·A-1 in Y,degree · h-1 in each Zi. • # coefficients = A K > |T| = # constraints ) nonzero solution • WLOG E(Y) doesn’t divide Q(Y,Z1,…,Zm).

  18. Proof of Expansion: Step 2 (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm: ForA=q-nmh, K=hm, |T|·AK-1)|LIST(T)|· K-1. Step 1: 9Q vanishing on T, deg ·A-1 in Y, h-1 in Zi, E-Q Step 2: Every f 2LIST(T) is a “root” of a related Q’. f(Y) 2LIST(T) ) 8 y2Fq Q(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y))= 0 ) Q(Y, f(Y), (fh mod E)(Y), …, (fhm-1 mod E)(Y))  0 ) Q(Y, f(Y), f(Y)h, …, f(Y)hm-1)  0 (mod E(Y)) ) Q’(f) = 0 in Fq[Y]/E(Y), whereQ’(Z) = Q(Y,Z,Zh,…,Zhm-1) mod E(Y)

  19. Proof of Expansion: Step 3 (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm: ForA=q-nmh, K=hm, |T|·AK-1)|LIST(T)|· K-1. Step 1: 9Q vanishing on T, deg ·A-1 in Y, h-1 in Zi, E-Q Step 2: 8 f2LIST(T)Q’(f) = 0 where Q’(Z) = Q(Y,Z,Zh,…,Zhm-1) mod E(Y) Step 3:Show that deg(Q’) · K-1 • Q’(Z) nonzerobecause Q(Y,Z1,….,Zm) not divisible by E(Y) & is of deg ·h-1 in Zi • deg(Q’(Z)) · h-1+(h-1)¢ h++(h-1)¢ hm-1 = hm-1 = K-1

  20. Proof of Expansion: Wrap-Up (f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) f = poly of degree ·n-1, E = irreducible of degree n Thm: ForA=q-nmh, K=hm, |T|·AK-1)|LIST(T)|· K-1. Step 1: 9Q vanishing on T, deg ·A-1 in Y, h-1 in Zi, E-Q Step 2: 8 f2LIST(T)Q’(f) = 0 where Q’(Z) = Q(Y,Z,Zh,…,Zhm-1) mod E(Y) Step 3:Show that deg(Q’) · K-1 Proof of Thm: |LIST(T)| · deg(Q’) · K-1. ¥

  21. S, |S| K D Our Result N M Thm: For every N, K, >0, 9explicit (K,A)expander with • degree D = poly(log N, 1/) • expansion A = (1-)¢D • #right vertices M = D2¢K1.01. |(S)|  A¢|S|

  22. Outline • Expander Construction • Application to Extractors • Connections • Conclusions

  23. Extractors: Original Motivation[SV84,Vaz85,VV85,CG85,Vaz87,CW89,Zuc90,Zuc91] • Randomization is pervasive in CS • Algorithm design, cryptography, distributed computing, … • Typically assume perfect random source. • Unbiased, independent random bits • Unrealistic? • Can we use a “weak” random source? • Source of biased & correlated bits. • More realistic model of physical sources. • (Randomness) Extractors: convert a weak random source into an almost-perfect random source.

  24. Applications of Extractors • Derandomization of (poly-time/log-space) algorithms [Sip88,NZ93,INW94, GZ97,RR99, MV99,STV99,GW02] • Distributed & Network Algorithms[WZ95,Zuc97,RZ98,Ind02]. • Hardness of Approximation [Zuc93,Uma99,MU01,Zuc06] • Data Structures [Ta02] • Cryptography [BBR85,HILL89,CDHKS00,Lu02,DRS04,NV04] • Metric Embeddings [Ind06]

  25. Extractors [NZ93] k-source of length n • Def: A (k,e)-extractor is Ext : {0,1}n£{0,1}d!{0,1}m s.t. 8k-source X, Ext(X,Ud) is e-close to Um. “seed” EXT drandom bits in variationdistance malmost-uniform bits 8x Pr[X=x]·2-k • Optimal (nonconstructive): d = log(n-k)+2log(1/)+O(1)m = k+d-2log(1/)-O(1)

  26. Our Result k-source of length n Thm: For every n, k, >0, 9 explicit (k,) extractor with seed length d=O(log(n/)) and output length m=.99k. • Previously achieved by [LRVW03] • Only worked for ¸ 1/no(1) • Complicated recursive construction “seed” EXT drandom bits malmost-uniform bits

  27. Approach: Condensers [RR99,RSW00] k-source of length n Def: A k! k’ condenser is Con : {0,1}n£{0,1}d!{0,1}m s.t. 8k-source X, Con(X,Ud) e-close to some k’-source. • Can extract from output: easier if k’/m > k/n. • Called lossless if k’=k+d. “seed” CON drandom bits ¼k’-source of length m

  28. Lossless Condensers  Expanders 2k {0,1}n x n-bit k-source Lemma [TUZ01]:Con :{0,1}n£{0,1}d!{0,1}m is a k! k+d condenser iff it defines a (2k,(1-)¢2d) expander. Proof ((): • Suffices to condense sources uniform on 2k strings. • Expansion ) can make 1-1 by moving  fraction of edges 2d CON y d-bit seed {0,1}m ¼ m-bit (k+d)-source Con(x,y) ¸ (1-) 2d¢ 2k

  29. Our Condenser Thm: For every N, K, >0, 9explicit (K,A)expander with • degree D = poly(log N, 1/) • expansion A = (1-)¢D • #right vertices M = D2¢K1.01.(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) Thm: For every n, k, >0, 9explicit k! k+d condenser w/ • seed length d = O(log n+log(1/)) • output length m=2d+1.01¢k • Con(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y))

  30. Our Extractor Condense:9explicit k! k+d condenser w/ • seed length d = O(log n+log(1/)) • output length m ¼ 1.01k • Con(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) Then Extract: apply extractor for min-entropy rate .99: • Constant • Ext(x,y) = y’th vertex on expander walk specified by x. • Extraction follows from Chernoff bound for expander walks [G98], via equivalence of extractors and samplers [Z96].

  31. Our Extractor Condense:9explicit k! k+d condenser w/ • seed length d = O(log n+log(1/)) • output length m ¼ 1.01k • Con(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) Then Extract: apply extractor for min-entropy rate .99: • Arbitrary : • Zuckerman’s extractor for constant min-entropy rate [Z96].

  32. Variations on the Condenser Thm: 9explicit k! k+d condenser w/ • seed length d = O(log n+log(1/)) • output length m ¼ 1.01k • Con(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) Variations (lose constant fraction of min-entropy): • “Repeated roots” [GS99] in analysis • seed length d = log n+log(1/)+O(1) • output length m = O(k ¢ log(n/))

  33. Variations on the Condenser Thm: 9explicit k! k+d condenser w/ • seed length d = O(log n+log(1/)) • output length m ¼ 1.01k • Con(f,y) =(y, f(y), (fh mod E)(y), …, (fhm-1 mod E)(y)) Variations (lose constant fraction of min-entropy): • E(Y) = Yq-1 - , for primitive root  [GR06]) (fhimod E)(y) = f (i y)) univariate analogue of Shaltiel-Umans extractor [SU01].

  34. Outline • Expander Construction • Application to Extractors • Connections • Conclusions

  35. Comparing List-Decoding Views  : {0,1}n£{0,1}d ! {0,1}d £{0,1}m T µ {0,1}d£ {0,1}m N=2n,D=2d,…

  36. Outline • Expander Construction • Application to Extractors • Connections • Conclusions

  37. Conclusions • List-decoding view ) best known constructions of • Highly unbalanced expanders • Lossless condensers • Randomness extractors • Push it further? • Nonbipartite expanders • Direct construction of extractor • Extractors optimal up to additive constants • Better list-decodable codes

More Related