Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T03:58:40.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Division in logspace-uniform NC 1

Published online by Cambridge University Press:  15 April 2002

Andrew Chiu ,
George Davida  and
Bruce Litow
Andrew Chiu
Affiliation:
EECS Department, University of Wisconsin-Milwaukee, Milwaukee, WI, U.S.A.
George Davida
Affiliation:
EECS Department, University of Wisconsin-Milwaukee, Milwaukee, WI, U.S.A.; e-mail: [email protected]
Bruce Litow
Affiliation:
School of Information Technology, James Cook University, Townsville, Qld. 4811, Australia; e-mail: [email protected]
Get access

Abstract

Beame, Cook and Hoover were the first to exhibita log-depth, polynomial size circuit family for integerdivision. However, the family was not logspace-uniform.In this paper we describe log-depth, polynomialsize, logspace-uniform, i.e., NC 1 circuit family forinteger division. In particular, by a well-known result this showsthat division is in logspace. We also refine the methodof the paper to show that division is in dlogtime-uniformNC 1.

Keywords

Parallel complexityNCinteger divisionuniformity.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 3 , May 2001 , pp. 259 - 275
Copyright
© EDP Sciences, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allender, E., Agrawal, M. and Datta, S., On TC0, AC0, and arithmetic circuits. J. Comput. System Sci. 60 (2000) 395-421.
E. Allender and D.A. Mix Barrington, Uniform circuits for division: Consequences and problems (2000) , .
Mix Barrington, D., Immerman, N. and Straubing, H., On uniformity within NC 1. J. Comput. System Sci. 41 (1990) 274-306. CrossRef
Beame, P., Cook, S. and Hoover, H., Log depth circuits for division and related problems. SIAM J. Comput. 15 (1986) 994-1003. CrossRef
Bertoni, A., Goldwurm, M. and Massazza, P., Counting problems and algebraic formal power series in noncommuting variables. Inform. Process. Lett. 34 (1990) 117-121. CrossRef
Borodin, A., On relating time and space to size and depth. SIAM J. Comput. 6 (1977) 733-744. CrossRef
Cook, S., A taxonomy of problems with fast parallel algorithms. Inform. and Control 64 (1985) 2-22. CrossRef
Davida, G. and Litow, B., Fast parallel arithmetic via modular representation. SIAM J. Comput. 20 (1991) 756-765. CrossRef
W. Hesse, Division is in uniform TC0. Comp. Sci., U. Mass. Amherst (2000).
Hitz, M.A. and Kaltofen, E., Integer division in residue number systems. IEEE Trans. Comput. 44 (1995) 983-989. CrossRef
N. Immerman, Descriptive Complexity. Springer-Verlag (1999).
D. Knuth, The Art of Computer Programming, Vol. 2. Addison-Wesley (1969).
Kruskal, C., Rudolph, L. and Snir, M., A complexity theory of efficient parallel algorithms. Theoret. Comput. Sci. 71 (1990) 95-132. CrossRef
B. Litow, Computing context-free grammar generating series. Inform. and Comput. (in press).
Macarie, I., Space-efficient deterministic simulation of probabilistic automata. SIAM J. Comput. 27 (1998) 448-465. CrossRef
Reif, J., Logarithmic depth circuits for algebraic functions. SIAM J. Comput. 15 (1986) 231-242. CrossRef
Ruzzo, W., On uniform circuit complexity. J. Comput. System Sci. 22 (1981) 365-383. CrossRef
R. Tanaka and N. Szabo, Residue Arithmetic and its Application to Computer Technology. McGraw-Hill (1968).
H. Vollmer, Introduction to Circuit Complexity. Springer-Verlag (1999).
I. Wegener, The Complexity of Boolean Functions. Wiley-Teubner (1987).