ECCC-Report TR13-086https://eccc.weizmann.ac.il/report/2013/086Comments and Revisions published for TR13-086en-usThu, 20 Jun 2013 00:07:09 +0300
Revision 1
| Pseudorandomness for Regular Branching Programs via Fourier Analysis |
Omer Reingold,
Thomas Steinke,
Salil Vadhan
https://eccc.weizmann.ac.il/report/2013/086#revision1We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, which follows as a special
case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of
$s^{1/2+o(1)}$ for arbitrary branching programs of size $s$). Our techniques also give seed length $n^{1/2+o(1)}$ for general oblivious, read-once branching programs of width $2^{n^{o(1)}}$, which is incomparable to the results of Impagliazzo et al.
Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $w$ has Fourier mass at most $(2w^2)^k$ at level $k$, independent of the length of the program. Thu, 20 Jun 2013 00:07:09 +0300https://eccc.weizmann.ac.il/report/2013/086#revision1
Paper TR13-086
| Pseudorandomness for Regular Branching Programs via Fourier Analysis |
Omer Reingold,
Thomas Steinke,
Salil Vadhan
https://eccc.weizmann.ac.il/report/2013/086We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, which follows as a special
case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of
$s^{1/2+o(1)}$ for arbitrary branching programs of size $s$). Our techniques also give seed length $n^{1/2+o(1)}$ for general oblivious, read-once branching programs of width $2^{n^{o(1)}}$, which is incomparable to the results of Impagliazzo et al.
Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $w$ has Fourier mass at most $(2w)^{2k}$ at level $k$, independent of the length of the program. Thu, 13 Jun 2013 07:07:54 +0300https://eccc.weizmann.ac.il/report/2013/086