The Parallel Evaluation of Arithmetic Expressions Without Division

As computers become capable of executing more arithmetic operations simultaneously, the question of compiling for such machines becomes more important. In this correspondence we consider arbitrary arithmetic expressions of n distinct variables with operations restricted to addition, subtraction, and multiplication. We first construct a scheme whereby any such expression can be evaluated in at most 3 log2 n + O(1) steps if sufficiently many processors are available. We then improve this result and reduce 3 log2 n to 2.465 log2 n. Finally, we deduce some results that apply when a fixed number of processors is available. CommentsOnly the Abstract is given here. The full paper appeared as [1] and was reprinted in [3, pages30–35]. For a more general result which allows division, and requires less processors, see [2]. References[1] R. P. Brent, D. J. Kuck and K. Maruyama, “The parallel evaluation of arithmetic expressions without division”,IEEE Transactions on Computers C-22 (1973), 532–534. MR 50#11843. rpb015.[2] R. P. Brent, “The parallel evaluation of general arithmetic expressions”, Journal of the ACM 21 (1974),201–206. CR 15#27055, MR 58#31996, Zbl 276.68010. rpb022.[3] R. P. Brent, Topics in computational complexity and the analysis of algorithms, Report TR-CS-80-14, DCS,ANU, October 1980, 375 pp. (D. Sc. thesis). rpb062. (R. P. Brent) Computer Centre, Australian National University, Canberra, Australia (D. J. Kuck and K. Maruyama) Department of Computer Science, University of Illinois, Urbana,Illinois 61801 1991 Mathematics Subject Classification. Primary 68Q22; Secondary 65Y05, 65Y20, 68M07, 68Q25.