# On the number of iterations required by Von Neumann addition

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2001)

- Volume: 35, Issue: 2, page 187-206
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topGrübel, Rudolf, and Reimers, Anke. "On the number of iterations required by Von Neumann addition." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 35.2 (2001): 187-206. <http://eudml.org/doc/92661>.

@article{Grübel2001,

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.},

author = {Grübel, Rudolf, Reimers, Anke},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {carry propagation; limit distributions; total variation distance; logarithmic periodicity; Gumbel distributions; discretization; large deviations; multiprecision arithmetic},

language = {eng},

number = {2},

pages = {187-206},

publisher = {EDP-Sciences},

title = {On the number of iterations required by Von Neumann addition},

url = {http://eudml.org/doc/92661},

volume = {35},

year = {2001},

}

TY - JOUR

AU - Grübel, Rudolf

AU - Reimers, Anke

TI - On the number of iterations required by Von Neumann addition

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2001

PB - EDP-Sciences

VL - 35

IS - 2

SP - 187

EP - 206

AB - 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.

LA - eng

KW - carry propagation; limit distributions; total variation distance; logarithmic periodicity; Gumbel distributions; discretization; large deviations; multiprecision arithmetic

UR - http://eudml.org/doc/92661

ER -

## References

top- [1] P. Billingsley, Probability and Measure, 2nd Ed. Wiley, New York (1986). Zbl0649.60001MR830424
- [2] 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). MR22442
- [3] P. Chassaing, J.F. Marckert and M. Yor, A stochastically quasi-optimal algorithm. Preprint (1999). Zbl1084.68145
- [4] V. Claus, Die mittlere Additionsdauer eines Paralleladdierwerks. Acta Inform. 2 (1973) 283-291. Zbl0304.68048MR366092
- [5] Th.H. Cormen, Ch.E. Leiserson and R.L. Rivest, Introduction to Algorithms. MIT Press, Cambridge, USA (1997). Zbl1047.68161MR1066870
- [6] Ph. Flajolet, X. Gourdon and Ph. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 (1995) 3-58. Zbl0869.68057MR1337752
- [7] O. Forster, Algorithmische Zahlentheorie. Vieweg, Braunschweig (1996). Zbl0870.11001
- [8] R. Grübel, Hoare’s selection algorithm: A Markov chain approach. J. Appl. Probab. 35 (1998) 36-45. Zbl0913.60059
- [9] R. Grübel, On the median-of-$k$ version of Hoare’s selection algorithm. RAIRO: Theoret. Informatics Appl. 33 (1999) 177-192. Zbl0946.68058
- [10] R. Grübel and U. Rösler, Asymptotic distribution theory for Hoare’s selection algorithm. Adv. Appl. Probab. 28 (1996) 252-269. Zbl0853.60033
- [11] D.E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching. Addison-Wesley, Reading (1973). Zbl0302.68010MR445948
- [12] D.E. Knuth, The average time for carry propagation. Nederl. Akad. Wetensch. Indag. Math. 40 (1978) 238-242. Zbl0382.10035MR497803
- [13] 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). Zbl0927.60027
- [14] C. McDiarmid and R.B. Hayward, Large deviations for Quicksort. J. Algorithms 21 (1996) 476-507. Zbl0863.68059MR1417660
- [15] M. Régnier, A limiting distribution for quicksort. RAIRO: Theoret. Informatics Appl. 23 (1989) 335-343. Zbl0677.68072MR1020478
- [16] S.I. Resnick, Extreme Values, Regular Variation and Point Processes. Springer, New York (1987). Zbl0633.60001MR900810
- [17] U. Rösler, A limit theorem for “Quicksort”. RAIRO: Theoret. Informatics Appl. 25 (1991) 85-100. Zbl0718.68026
- [18] W. Rudin, Real and Complex Analysis, 2nd Ed. Tata McGraw-Hill, New Delhi (1974). Zbl0278.26001MR344043
- [19] N.R. Scott, Computer Number Systems & Arithmetic. Prentice-Hall, New Jersey (1985). Zbl0613.65046
- [20] R. Sedgewick and Ph. Flajolet, An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading (1996). Zbl0841.68059
- [21] I. Wegener, Effiziente Algorithmen für grundlegende Funktionen. B.G. Teubner, Stuttgart (1996). Zbl0697.68011MR1076624