Topics in Computational Complexity and the Analysis of Algorithms

TOPICS IN COMPUTATIONAL COMPLEXITY AND THE ANALYSIS OF ALGORITHMS

RICHARD P. BRENT

Abstract A collection of 18 papers by R. P. Brent and various co-authors, on the following topics: 1. Parallel evaluation of arithmetic expressions. 2. Circuits for arithmetic operations. 3. Continuous models for discrete algorithms. 4. Algorithms for manipulating formal power series. 5. The complexity of algorithms for solving nonlinear equations. 6. Asymptotically fast algorithms for high-precision computations. The theme of the collection is the derivation of rigorous bounds on the cost of certain computations. Cost may be measured in several diﬀerent ways. For example, the number of arithmetic operations performed on integers or real numbers, the number of Boolean operations, the number of function and derivative evaluations, the time required for the computation on a parallel machine, or the area-time product for a VLSI circuit. Most of the papers are concerned with upper bounds, which are established by exhibiting an algorithm and analysing its performance. However, papers [6, 17] establish nontrivial lower bounds.

Comments Only an Abstract is given here. The full collection appeared as [19] and was submitted to Monash University as a D.Sc. thesis. The papers included are [1–18] below.

References [1] R. P. Brent, “On the addition of binary numbers”, IEEE Transactions on Computers C-19 (1970), 758–759. CR 12#20898. rpb003. [2] R. P. Brent, “The computational complexity of iterative methods for systems of nonlinear equations”, in Complexity of Computer Computations (edited by R. E. Miller and J. W. Thatcher), Plenum Press, New York, 1972, 61–71. MR 51#9575, 52#4703. rpb012. [3] R. P. Brent, “Reducing the retrieval time of scatter storage techniques”, Communications of the ACM 16 (1973), 105–109. Zbl 251.68019. See also “Comment on Brent’s scatter storage algorithm”, ibid, 703. rpb013.

1991 Mathematics Subject Classiﬁcation. Primary 68Q25; Secondary 65Y20, 65Y05, 65H05, 65H10, 65G05, 65-06, 68-06. Key words and phrases. Arithmetic circuits, analysis of algorithms, computational complexity, parallel computation, arithmetic expressions, power series, composition, reversion, nonlinear equations, greatest common divisors, high-precision computation, VSLI layouts, binary addition, binary multiplication, parallel preﬁx, hash coding, scatter storage, elementary function evaluation. Thanks to my co-authors D. J. Kuck, H. T. Kung, K. Maruyama, J. F. Traub, S. Winograd, and P. Wolfe, for their invaluable contributions. This work includes eighteen papers which were originally published elsewhere and for which copyright is held by the original publishers or authors. Details are given on the title page preceeding each paper. Introduction and Bibliography c 1980, R. P. Brent. Abstract and Comments c 1993, R. P. Brent. rpb062a typeset using AMS-LATEX. 2 RPB062A, BRENT ET AL

[4] R. P. Brent, “Some efficient algorithms for solving systems of nonlinear equations”, SIAM J. Numerical Anal- ysis 10 (1973), 327–344 (George E. Forsythe memorial issue). MR 48#10096, CR 17#29965, Zbl 258.65051. rpb014. [5] 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. [6] R. P. Brent, S. Winograd and P. Wolfe, “Optimal iterative processes for rootfinding”, Numerische Mathematik 20 (1973), 327–341. CR 15#26753, MR 47#6079. rpb016. [7] R. P. Brent, “The parallel evaluation of arithmetic expressions in logarithmic time”, in Complexity of Sequen- tial and Parallel Numerical Algorithms (edited by J. F. Traub), Academic Press, New York, 1973, 83–102. CR 15#26540, 15#27335; MR 50#15432; Zbl 273.68037. rpb018. [8] R. P. Brent, “The parallel evaluation of general arithmetic expressions”, J. ACM 21 (1974), 201–206. CR 15#27055, MR 58#31996, Zbl 276.68010. rpb022. [9] R. P. Brent, “A class of optimal-order zero-finding methods using derivative evaluations”, in Analytic Com- putational Complexity (edited by J. F. Traub), Academic Press, 1975, 59–73. MR 52#15938, 54#9073; Zbl 342.65030. rpb027. [10] R. P. Brent, “Multiple-precision zero-finding methods and the complexity of elementary function evaluation”, in Analytic Computational Complexity (edited by J. F. Traub), Academic Press, New York, 1975, 151–176. MR 52#15938, 54#11843; Zbl 342.65031. rpb028. [11] R. P. Brent, “The complexity of multiple-precision arithmetic”, in The Complexity of Computational Problem Solving (edited by R. S. Anderssen and R. P. Brent), University of Queensland Press, Brisbane, 1976, 126–165. rpb032. [12] R. P. Brent, “Fast multiple–precision evaluation of elementary functions”, J. ACM 23 (1976), 242–251. MR 52#16111, Zbl 324.65018. Also appeared as Report TR STAN-CS-75-515, Department of Computer Science, Stanford University, August 1975, 22 pp. rpb034. [13] R. P. Brent, “Analysis of the binary Euclidean algorithm”, in New Directions and Recent Results in Algorithms and Complexity (edited by J. F. Traub), Academic Press, New York, 1976, 321–355. MR 54#14417, 55#11701; Zbl 363.00013, 373.68040. rpb037. [14] R. P. Brent and H. T. Kung, “Fast algorithms for composition and reversion of multivariate power series”, in Proceedings of a Conference on Theoretical Computer Science (held at the University of Waterloo), Depart- ment of Computer Science, University of Waterloo, Waterloo, Ontario (August 1977), 149–158. Zbl 404.00019, 411.68043. rpb039. [15] R. P. Brent and H. T. Kung, “Fast algorithms for manipulating formal power series”, J. ACM 25 (1978), 581–595. CR 20#34535, MR 58#25090, Zbl 388.68052. rpb045. [16] R. P. Brent and J. F. Traub, “On the complexity of composition and generalized composition of power series”, SIAM J. Computing 9 (1980), 54–66. MR 81b:68042. rpb050. [17] R. P. Brent and H. T. Kung, “The area-time complexity of binary multiplication”, J. ACM 28 (1981), 521–534. CR 22#38242, MR 82i:68027. Corrigendum: ibid 29 (1982), 904. MR 83j:68046. rpb055. [18] R. P. Brent and H. T. Kung, “A regular layout for parallel adders”, IEEE Transactions on Computers C-31 (1982), 260–264. MR 83b:68002. rpb060. [19] 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.

Computer Science Department, Australian National University, Canberra E-mail address: [email protected]