Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma, Umans, and Zuckerman (STOC ‘01) required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy (FOCS ‘05). Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC ‘03) and improving upon it when the error parameter is small (e.g. 1/poly(n)).

[1]  M. Murty.  Ramanujan Graphs , 1965 .

[2]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[3]  Victor Shoup,et al.  New algorithms for finding irreducible polynomials over finite fields , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[4]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[5]  A. Wigderson,et al.  Disperser graphs, deterministic amplification, and imperfect random sources (גרפים מפזרים, הגברה דטרמיניסטית ומקורות אקראים חלשים.) , 1991 .

[6]  Leonid A. Levin,et al.  Pseudo-random Generation from one-way functions (Extended Abstracts) , 1989, STOC 1989.

[7]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[8]  Avi Wigderson,et al.  Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications , 1993, Comb..

[9]  H. Niederreiter,et al.  Introduction to finite fields and their applications: Factorization of Polynomials , 1994 .

[10]  Aravind Srinivasan,et al.  Computing with very weak random sources , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[11]  Avi Wigderson,et al.  Tiny Families of Functions with Random Properties: A Quality-Size Trade-off for Hashing , 1997, Electron. Colloquium Comput. Complex..

[12]  Nabil Kahale,et al.  Eigenvalues and expansion of regular graphs , 1995, JACM.

[13]  Noam Nisan,et al.  Randomness is Linear in Space , 1996, J. Comput. Syst. Sci..

[14]  D. Zuckerman,et al.  Randomness-optimal oblivious sampling , 1997, Random Struct. Algorithms.

[15]  Madhu Sudan,et al.  Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[16]  D. Gillman.  A Chernoff bound for random walks on expander graphs , 1998, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[17]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[18]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[19]  Noam Nisan,et al.  Extracting Randomness: A Survey and New Constructions , 1999, J. Comput. Syst. Sci..

[20]  Ran Raz,et al.  Extracting all the randomness and reducing the error in Trevisan's extractors , 1999, STOC '99.

[21]  Avi Wigderson,et al.  Extracting randomness via repeated condensing , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[22]  Jaikumar Radhakrishnan,et al.  Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators , 2000, SIAM J. Discret. Math..

[23]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[24]  Avi Wigderson,et al.  Extractors and pseudo-random generators with optimal seed length , 2000, STOC '00.

[25]  Peter Bro Miltersen,et al.  Are bitvectors optimal? , 2000, STOC '00.

[26]  Omer Reingold,et al.  Randomness Conductors and Constant-Degree Expansion Beyond the Degree / 2 Barrier , 2001 .

[27]  Amnon Ta-Shma,et al.  Loss-less condensers, unbalanced expanders, and extractors , 2001, STOC '01.

[28]  Luca Trevisan,et al.  Extractors and pseudorandom generators , 2001, JACM.

[29]  Amnon Ta-Shma,et al.  Extractors from Reed-Muller Codes , 2001, Electron. Colloquium Comput. Complex..

[30]  Christopher Umans.  Pseudo-random generators for all hardnesses , 2002, STOC '02.

[31]  Amnon Ta-Shma,et al.  Storing information with extractors , 2002, Inf. Process. Lett..

[32]  L. Fortnow,et al.  Recent Developments in Explicit Constructions of Extractors , 2002, Bull. EATCS.

[33]  Venkatesan Guruswami,et al.  Combinatorial bounds for list decoding , 2002, IEEE Trans. Inf. Theory.

[34]  Avi Wigderson,et al.  Extractors: optimal up to constant factors , 2003, STOC '03.

[35]  Amnon Ta-Shma,et al.  Extractor codes , 2001, IEEE Transactions on Information Theory.

[36]  David Zuckerman.  Simulating BPP using a general weak random source , 2005, Algorithmica.

[37]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[38]  Christopher Umans,et al.  Simple extractors for all min-entropies and a new pseudorandom generator , 2005, JACM.

[39]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[40]  Amnon Ta-Shma,et al.  Better lossless condensers through derandomized curve samplers , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[41]  Prasad Tetali,et al.  Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs , 2006, Random Struct. Algorithms.

[42]  Christopher Umans,et al.  On Obtaining Pseudorandomness from Error-Correcting Codes , 2006, FSTTCS.

[43]  Venkatesan Guruswami,et al.  Extractors and condensers from univariate polynomials , 2006, Electron. Colloquium Comput. Complex..

[44]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[45]  Venkatesan Guruswami,et al.  Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes , 2007, JACM.

[46]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[47]  David Zuckerman,et al.  Extractors for Three Uneven-Length Sources , 2008, APPROX-RANDOM.