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Practically efficient methods for performing bit-reversed permutation in C++11 on the x86-64 architecture

Christian Knauth, Boran Adas, Daniel Whitfield, Xuesong Wang, Lydia Ickler, Tim Conrad, Oliver Serang
Freie Universitat Berlin, Institut fur Informatik
arXiv:1708.01873 [cs.MS], (2 Aug 2017)

@article{knauth2017practically,

   title={Practically efficient methods for performing bit-reversed permutation in C++11 on the x86-64 architecture},

   author={Knauth, Christian and Adas, Boran and Whitfield, Daniel and Wang, Xuesong and Ickler, Lydia and Conrad, Tim and Serang, Oliver},

   year={2017},

   month={aug},

   archivePrefix={"arXiv"},

   primaryClass={cs.MS}

}

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The bit-reversed permutation is a famous task in signal processing and is key to efficient implementation of the fast Fourier transform. This paper presents optimized C++11 implementations of five extant methods for computing the bit-reversed permutation: Stockham auto-sort, naive bitwise swapping, swapping via a table of reversed bytes, local pairwise swapping of bits, and swapping via a cache-localized matrix buffer. Three new strategies for performing the bit-reversed permutation in C++11 are proposed: an inductive method using the bitwise XOR operation, a template-recursive closed form, and a cache-oblivious template-recursive approach, which reduces the bit-reversed permutation to smaller bit-reversed permutations and a square matrix transposition. These new methods are compared to the extant approaches in terms of theoretical runtime, empirical compile time, and empirical runtime. The template-recursive cache-oblivious method is shown to be competitive with the fastest known method; however, we demonstrate that the cache-oblivious method can more readily benefit from parallelization on multiple cores and on the GPU.
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