Error reduction for weighted PRGs against read once branching programs

Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [5], are a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered, rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and Liao [6] somewhat simplified the technically involved BCG construction, also obtaining some improvement in parameters. In this work we devise an error reduction procedure for PRGs against ROBPs. More precisely, our procedure transforms any PRG against length n width w ROBP with error 1/poly(n) having seed length s to a WPRG with seed length s + O(log w/ε · log log 1/ε). By instantiating our procedure with Nisan's PRG [17] we obtain a WPRG with seed length O(log n · log(nw) + log w/ε · log log 1/ε). This improves upon [5] and is incomparable with [6]. Our construction is significantly simpler on the technical side and is conceptually cleaner. Another advantage of our construction is its low space complexity O(log nw) + poly(log log 1/ε) which is logarithmic in n for interesting values of the error parameter ε. Previous constructions (like [5, 6]) specify the seed length but not the space complexity, though it is plausible they can also achieve such (or close) space complexity.

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

[2]  Michael Saks,et al.  BP H SPACE(S)⊆DSPACE(S 3/2 ) , 1999, FOCS 1999.

[3]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[4]  Richard Peng,et al.  Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs , 2016, STOC.

[5]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[6]  Eshan Chattopadhyay,et al.  Optimal error pseudodistributions for read-once branching programs , 2020, Electron. Colloquium Comput. Complex..

[7]  Allan Borodin,et al.  Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines , 1984, Inf. Control..

[8]  Gil Cohen,et al.  Pseudorandom Pseudo-distributions with Near-Optimal Error for Read-Once Branching Programs , 2020, SIAM J. Comput..

[9]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[10]  A. Meyer.  Review: Walter J. Savitch, Relationships between Nondeterministic and Deterministic Tape Complexities , 1974 .

[11]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[12]  Avi Wigderson,et al.  In search of an easy witness: exponential time vs. probabilistic polynomial time , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[13]  Oded Goldreich,et al.  Computational complexity: a conceptual perspective , 2008, SIGA.

[14]  Richard Peng,et al.  Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[15]  Noam Nisan,et al.  Pseudorandomness for network algorithms , 1994, STOC '94.

[16]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[17]  Emanuele Viola,et al.  Constant-Depth Circuits for Arithmetic in Finite Fields of Characteristic Two , 2006, STACS.

[18]  Kuan Cheng,et al.  Hitting sets give two-sided derandomization of small space , 2020, Electron. Colloquium Comput. Complex..

[19]  Ran Raz,et al.  On recycling the randomness of states in space bounded computation , 1999, STOC '99.

[20]  David Zuckerman,et al.  Simple Optimal Hitting Sets for Small-Success RL , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[21]  Salil Vadhan,et al.  High-precision Estimation of Random Walks in Small Space , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).