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2021, no. 19 (2021).
A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a
pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ordered branching programs whose seed length has a better dependence on the error parameter
than the classic PRG construction of Nisan (STOC 1990 and Combinatorica 1992).
In this work, we give an explicit construction of PRPGs that achieve parameters that are
impossible to achieve by a PRG. In particular, we construct a PRPG for
ordered permutation branching programs of unbounded width with a single accept state that has seed length
\(\tilde{O}(\log^{3/2}n)\) for error parameter
\( \epsilon = 1/ \mathrm{poly}(n)\), where
\(n\) is the input length. In contrast, recent work of Hoza et al. (ITCS 2021) shows that any PRG for this model requires seed length
\( \Omega(\log^2n)\) to achieve error
\( \epsilon = 1/ \mathrm{poly}(n)\).
As a corollary, we obtain explicit PRPGs with seed length
\(\tilde{O}(\log^{3/2}n)\) and error
\( \epsilon = 1/ \mathrm{poly}(n)\) for ordered permutation branching programs of width
\(w = \mathrm{poly}(n) \)with an arbitrary number of accept states. Previously, seed length
\(o(\log^2n)\) was only known when both the width and the reciprocal of the error are subpolynomial, i.e.
\(w= n^{o(1)} \) and
\(\epsilon = 1/n^{o(1)}\)(Braverman, Rao, Raz, Yehudayoff, FOCS 2010 and SICOMP 2014).
The starting point for our results are the recent space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadenijad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS 2020), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving PRPGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (personal communication, January 2021).