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Expander Graphs, Randomness Extractors and List-Decodable Codes Salil Vadhan Harvard University Joint work with Venkat Guruswami (UW) & Chris Umans (Caltech)

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[GW94,WZ95,TUZ01, RVW00,CRVW02] Connections in Pseudorandomness Randomness Extractors Expander Graphs List-Decodable Error-Correcting Codes Pseudorandom Generators Samplers [Tre99,RRV99, ISW99,SU01,U02] [Tre99,TZ01, TZS01,SU01] [CW89,Z96] This Work [PV05,GR06] This Work

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Outline Expander Construction Application to Extractors Connections Conclusions

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(Bipartite) Expander Graphs Goals: Minimize D Maximize A Minimize M | (S)| A ¢ |S| D N M S, |S| K Nonconstructive: D = O(log(N/M)/ ) A = (1- ) ¢ D M = (KD/ “ (K,A) expander” O(1) if M=N log N) if M ·p N =

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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)).

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Advantage of Expansion (1- ) ¢ D At least (1-2 ) D |S| elements of (S) are unique neighbors: touch exactly one edge from S | (S)| (1- ) D |S| D N M S, |S| K x Fault tolerance: Even if an adversary removes most (say ¾) edges from each vertex, lossless expansion maintained (with =4 )

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Application to Data Structures [BMRS00] Goal: store small S ½ [N] s.t. can test membership by (probabilistically) reading 1 bit. Expansion (1- ) ¢ D ) 9 0,1 assignment to [M] s.t. for every x 2 [N], a 1-O( ) fraction of neighbors have correct answer! D N M S, |S| K | (S)| (1- ) ¢ D ¢ |S| /2 0 0 0 1 1 0 1 1 0

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Application to Data Structures [BMRS00] 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 D N M S, |S| K /2 0 0 0 1 1 0 1 1 0

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Explicit Constructions Nonconstructive O(log(N/M)) (1- ) ¢ DO(KD Ramanujan graphs […LPS86,M88] O(1) ¼ D/2 [Kah94] N Zig-zag CRVW02] O(1) (1- ) ¢ D N Ta-Shma, Umans, Zuckerman [TUZ01] polylog(N) quasipoly(log N) (1- ) ¢ D quasipoly(KD) poly(KD) Our Result polylog(N) (1- ) ¢ D poly(KD) degree D expansion A |right-side| M arbitrary constant. quasipoly(t)=exp(polylog t)

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Our Result Thm: For every N, K, >0, 9 explicit (K,A) expander with degree D = poly(log N, 1/ ) expansion A = (1- ) ¢ D #right vertices M = D 2 ¢ K 1.01. | (S)| A ¢ |S| D N M S, |S| K

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Our Construction Left vertices = F q n = polys of degree · n-1 over F q Degree = q Right vertices = F q m+1 ( f,y ) = y ’th neighbor of f = (y, f(y), (f h mod E)(y), (f h 2 mod E)(y), …, (f h m-1 mod E)(y)) where E(Y) = irreducible poly of degree n h = a parameter Thm: This is a (K,A) expander with K=h m, A = q-hnm.

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Setting Parameters ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n N = F q n, D = q, M = F q m+1 Thm: This is a (K,A) expander with K=h m, A = q-hnm. Set h = poly ( nm / ) q = h 1.01 Then: D = q = poly(log N, 1/ ) A = q-hnm ¸ (1- ) ¢ D M = q m+1 = q ¢ (h 1.01 ) m = D ¢ K 1.01

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

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List-Decoding View of Expanders For T µ [M], define LIST(T) = {x 2 [N] : (x) µ T} Lemma: G is a (=K,A) expander iff for all T µ [M] of size AK-1, we have |LIST(T)| · K-1 | (S)| A ¢ K D N S, |S|=K M “ (=K,A) expander”

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Comparing List-Decoding Views : [N] £ [D] ! [D] £ [M] T µ [D] £ [M] ObjectInterpretation x 2 LIST(T) iffDecoding Problem expanders x,y) = y ’th nbr of x 8 y (x,y) 2 T |T| < AK ) |LIST(T)| < K list-decodable codes x,y) = (y,ECC(x) y ) Pr y [ (x,y) 2 T] ¸ 1/M + T = {(y,r y )} ) |LIST(T)| < K

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Proof of Expansion ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n Thm: For A=q-nmh and any K · h m, we have T µ F q m+1 of size AK-1 ) |LIST(T)| · K-1 Proof Outline (following [S97,GS99,PV05] ): 1.Find a low-degree poly Q vanishing on T. 2.Show that every f 2 LIST(T) is a “root” of a related polynomial Q’. 3.Show that deg(Q’) · K-1 =

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Proof of Expansion: Step 1 ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n Thm: For A=q-nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. Step 1: Find a low-degree poly Q vanishing on T. Take Q(Y,Z 1,…,Z m ) to be of degree · A-1 in Y, degree · h-1 in each Z i. # coefficients = A K > |T| = # constraints ) nonzero solution WLOG E(Y) doesn’t divide Q(Y,Z 1,…,Z m ).

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Proof of Expansion: Step 2 ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n Thm: For A=q-nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. Step 1: 9 Q vanishing on T, deg · A-1 in Y, h-1 in Z i, E - Q Step 2: Every f 2 LIST(T) is a “root” of a related Q’. f(Y) 2 LIST(T) ) 8 y 2 F q Q(y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) = 0 ) Q(Y, f(Y), (f h mod E)(Y), …, (f h m-1 mod E)(Y)) 0 ) Q(Y, f(Y), f(Y) h, …, f(Y) h m-1 ) 0 (mod E(Y)) ) Q’(f) = 0 in F q [Y]/E(Y), where Q’(Z) = Q(Y,Z,Z h,…,Z h m-1 ) mod E(Y)

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Proof of Expansion: Step 3 ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n Thm: For A=q-nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. Step 1: 9 Q vanishing on T, deg · A-1 in Y, h-1 in Z i, E - Q Step 2: 8 f 2 LIST(T) Q’(f) = 0 where Q’(Z) = Q(Y,Z,Z h,…,Z h m-1 ) mod E(Y) Step 3: Show that deg(Q’) · K-1 Q’(Z) nonzero because Q(Y,Z 1,….,Z m ) not divisible by E(Y) & is of deg · h-1 in Z i deg(Q’(Z)) · h-1+(h-1) ¢ h+ +(h-1) ¢ h m-1 = h m -1 = K-1

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Proof of Expansion: Wrap-Up ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, E = irreducible of degree n Thm: For A=q-nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. Step 1: 9 Q vanishing on T, deg · A-1 in Y, h-1 in Z i, E - Q Step 2: 8 f 2 LIST(T) Q’(f) = 0 where Q’(Z) = Q(Y,Z,Z h,…,Z h m-1 ) mod E(Y) Step 3: Show that deg(Q’) · K-1 Proof of Thm: |LIST(T)| · deg(Q’) · K-1. ¥

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Our Result Thm: For every N, K, >0, 9 explicit (K,A) expander with degree D = poly(log N, 1/ ) expansion A = (1- ) ¢ D #right vertices M = D 2 ¢ K 1.01. | (S)| A ¢ |S| D N M S, |S| K

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Outline Expander Construction Application to Extractors Connections Conclusions

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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.

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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]

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Def: A (k, ) -extractor is Ext : {0,1} n £ {0,1} d ! {0,1} m s.t. 8 k -source X, Ext ( X,U d ) is -close to U m. 8 x Pr [ X = x ] · 2 -k Extractors [NZ93] d random bits “seed” Optimal (nonconstructive): d = log(n-k)+2log(1/ )+O(1) m = k+d-2log(1/ )-O(1) E XT k - source of length n m almost-uniform bits in variation distance

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Our Result d random bits “seed” E XT k - source of length n m almost-uniform bits 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/n o(1) –Complicated recursive construction

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

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2k2k Lossless Condensers Expanders Lemma [TUZ01]: Con : {0,1} n £ {0,1} d ! {0,1} m is a k ! k+d condenser iff it defines a (2 k,(1- ) ¢ 2 d ) expander. Proof ( ( ): Suffices to condense sources uniform on 2 k strings. Expansion ) can make 1-1 by moving fraction of edges {0,1} n {0,1} m 2d2d ¸ (1- ) 2 d ¢ 2 k n - bit k - source ¼ m -bit ( k+d) - source d -bit seed C ON x Con(x,y) y

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Our Condenser Thm: For every N, K, >0, 9 explicit (K,A) expander with degree D = poly(log N, 1/ ) expansion A = (1- ) ¢ D #right vertices M = D 2 ¢ K 1.01. ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) Thm: For every n, k, >0, 9 explicit 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), (f h mod E)(y), …, (f h m-1 mod E)(y))

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Our Extractor Condense: 9 explicit 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), (f h mod E)(y), …, (f h m-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].

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Our Extractor Condense: 9 explicit 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), (f h mod E)(y), …, (f h m-1 mod E)(y)) Then Extract: apply extractor for min-entropy rate.99: Arbitrary –Zuckerman’s extractor for constant min-entropy rate [Z96].

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Variations on the Condenser Thm: 9 explicit 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), (f h mod E)(y), …, (f h m-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/ ))

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Variations on the Condenser Thm: 9 explicit 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), (f h mod E)(y), …, (f h m-1 mod E)(y)) Variations (lose constant fraction of min-entropy): E(Y) = Y q-1 - , for primitive root [GR06] ) (f h i mod E)(y) = f ( i y) ) univariate analogue of Shaltiel-Umans extractor [SU01].

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Outline Expander Construction Application to Extractors Connections Conclusions

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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=2 n,D=2 d,… ObjectInterpretation x 2 LIST(T) iffDecoding Problem expanders x,y) = y ’th nbr of x 8 y (x,y) 2 T |T| < AK ) |LIST(T)| < K list-decodable codes x,y) = (y,ECC(x) y ) Pr y [ (x,y) 2 T] ¸ 1/2 m + T = {(y,r y )} ) |LIST(T)| < K extractors Pr y [ (x,y) 2 T] ¸ |T|/2 m+d + 8 T |LIST(T)| < K lossy condensers Pr y [ (x,y) 2 T] ¸ |T|/2 m+d + |T| · K’-1 ) |LIST(T)| · K

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Outline Expander Construction Application to Extractors Connections Conclusions

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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

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