Towards an Optimal Bit-Reversal Permutation Program

The speed of many computations is limited not by the number of arithmetic operations but by the time it takes to move and rearrange data in the increasingly complicated memory hierarchies of modern computers. Array transpose and the bit-reversal permutation – trivial operations on a RAM – present non-trivial problems when designing highly-tuned scientific library functions, particular for the Fast Fourier Transform. We prove a precise bound for RoCol, a simple pebble-type game that is relevant to implementing these permutations. We use RoCol to give lower bounds on the amount of memory traffic in a computer with four-levels of memory (registers, cache, TLB, and memory), taking into account such “messy” features as block moves and set-associative caches. The insights from this analysis lead to a bit-reversal algorithm whose performance is close to the theoretical minimum. Experiments show it performs significantly better than every program in a comprehensive study of 30 published algorithms. 1. Background and related work Given binary strings a and b, let ab denote their concatenation and r(a) denote the reversal of a. Copyright 1998 IEEE. Published in the Proceedings of FOCS’98, 8-11 November 1998 in Palo Alto, CA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ 088551331, USA. Telephone: + Intl. 732-562-3966. Thus, for instance, r(01101) = 10110, and r(ab) = r(b)r(a). Arrays will be indexed by binary strings. The pseudocode statement “for i = 0 to N-1” means that i iterates through all binary strings of length lg(N), where lg represents log base two. Consider the following three programs, where N is a power of 2, N = N1 N2, and A and B are arrays of length N: