Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T23:42:14.622Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of iterations required by Von Neumann addition

Published online by Cambridge University Press:  15 April 2002

Rudolf Grübel  and
Anke Reimers
Rudolf Grübel
Affiliation:
Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, 30060 Hannover, Germany; (rgrubel@stochastik.uni-hannover.de)
Anke Reimers
Affiliation:
Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, 30060 Hannover, Germany
Get access

Abstract

We investigate the number of iterations needed by an addition algorithm due to Burks et al. if the input is random. Several authors have obtained results on the average case behaviour, mainly using analytic techniques based on generating functions. Here we take a more probabilistic view which leads to a limit theorem for the distribution of the random number of steps required by the algorithm and also helps to explain the limiting logarithmic periodicity as a simple discretization phenomenon.

Keywords

Carry propagationlimit distributionstotal variation distancelogarithmic periodicityGumbel distributionsdiscretizationlarge deviations.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 2 , March 2001 , pp. 187 - 206
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

P. Billingsley, Probability and Measure, 2nd Ed. Wiley, New York (1986).
A.W. Burks, H.H. Goldstine and J. von Neumann, Preliminary discussion of the logical design of an electronic computing instrument. Inst. for Advanced Study Report (1946). Reprinted in John von Neumann Collected Works, Vol. 5. Pergamon Press, New York (1961).
P. Chassaing, J.F. Marckert and M. Yor, A stochastically quasi-optimal algorithm. Preprint (1999).
Claus, V., Die mittlere Additionsdauer eines Paralleladdierwerks. Acta Inform. 2 (1973) 283-291. CrossRef
Th.H. Cormen, Ch.E. Leiserson and R.L. Rivest, Introduction to Algorithms. MIT Press, Cambridge, USA (1997).
Ph. Flajolet, X. Gourdon, Ph. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 (1995) 3-58. CrossRef
O. Forster, Algorithmische Zahlentheorie. Vieweg, Braunschweig (1996).
Grübel, R., Hoare's selection algorithm: A Markov chain approach. J. Appl. Probab. 35 (1998) 36-45. CrossRef
Grübel, R., On the median-of-k version of Hoare's selection algorithm. RAIRO: Theoret. Informatics Appl. 33 (1999) 177-192.
Grübel, R. and Rösler, U., Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Probab. 28 (1996) 252-269. CrossRef
D.E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching. Addison-Wesley, Reading (1973).
Knuth, D.E., The average time for carry propagation. Nederl. Akad. Wetensch. Indag. Math. 40 (1978) 238-242. CrossRef
C. McDiarmid, Concentration, in Probabilistic Methods for Algorithmic Discrete Mathematics, edited by M. Habib, C. McDiarmid, J. Ramirez-Alfonsin and B. Reed. Springer, Berlin (1998).
McDiarmid, C. and Hayward, R.B., Large deviations for Quicksort. J. Algorithms 21 (1996) 476-507. CrossRef
Régnier, M., A limiting distribution for quicksort. RAIRO: Theoret. Informatics Appl. 23 (1989) 335-343.
S.I. Resnick, Extreme Values, Regular Variation and Point Processes. Springer, New York (1987).
Rösler, U., A limit theorem for ``Quicksort". RAIRO: Theoret. Informatics Appl. 25 (1991) 85-100.
W. Rudin, Real and Complex Analysis, 2nd Ed. Tata McGraw-Hill, New Delhi (1974).
N.R. Scott, Computer Number Systems & Arithmetic. Prentice-Hall, New Jersey (1985).
R. Sedgewick and Ph. Flajolet, An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading (1996).
I. Wegener, Effiziente Algorithmen für grundlegende Funktionen. B.G. Teubner, Stuttgart (1996).