On the number of iterations required by Von Neumann addition
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 2, pp. 187-206.

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.

Classification : 68Q25,  65Y20
Mots clés : carry propagation, limit distributions, total variation distance, logarithmic periodicity, Gumbel distributions, discretization, large deviations
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author = {Gr\"ubel, Rudolf and Reimers, Anke},
title = {On the number of iterations required by {Von} {Neumann} addition},
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Grübel, Rudolf; Reimers, Anke. On the number of iterations required by Von Neumann addition. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 2, pp. 187-206. http://www.numdam.org/item/ITA_2001__35_2_187_0/

[1] P. Billingsley, Probability and Measure, 2nd Ed. Wiley, New York (1986). | MR 830424 | Zbl 0649.60001

[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). | MR 22442

[3] P. Chassaing, J.F. Marckert and M. Yor, A stochastically quasi-optimal algorithm. Preprint (1999).

[4] V. Claus, Die mittlere Additionsdauer eines Paralleladdierwerks. Acta Inform. 2 (1973) 283-291. | MR 366092 | Zbl 0304.68048

[5] Th.H. Cormen, Ch.E. Leiserson and R.L. Rivest, Introduction to Algorithms. MIT Press, Cambridge, USA (1997). | MR 1066870 | Zbl 1047.68161

[6] Ph. Flajolet, X. Gourdon and Ph. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 (1995) 3-58. | MR 1337752 | Zbl 0869.68057

[7] O. Forster, Algorithmische Zahlentheorie. Vieweg, Braunschweig (1996). | Zbl 0870.11001

[8] R. Grübel, Hoare's selection algorithm: A Markov chain approach. J. Appl. Probab. 35 (1998) 36-45. | Zbl 0913.60059

[9] R. Grübel, On the median-of-$k$ version of Hoare’s selection algorithm. RAIRO: Theoret. Informatics Appl. 33 (1999) 177-192. | Numdam | Zbl 0946.68058

[10] R. Grübel and U. Rösler, Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Probab. 28 (1996) 252-269. | Zbl 0853.60033

[11] D.E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching. Addison-Wesley, Reading (1973). | MR 445948 | Zbl 0302.68010

[12] D.E. Knuth, The average time for carry propagation. Nederl. Akad. Wetensch. Indag. Math. 40 (1978) 238-242. | MR 497803 | Zbl 0382.10035

[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). | Zbl 0927.60027

[14] C. Mcdiarmid and R.B. Hayward, Large deviations for Quicksort. J. Algorithms 21 (1996) 476-507. | MR 1417660 | Zbl 0863.68059

[15] M. Régnier, A limiting distribution for quicksort. RAIRO: Theoret. Informatics Appl. 23 (1989) 335-343. | Numdam | MR 1020478 | Zbl 0677.68072

[16] S.I. Resnick, Extreme Values, Regular Variation and Point Processes. Springer, New York (1987). | MR 900810 | Zbl 0633.60001

[17] U. Rösler, A limit theorem for “Quicksort”. RAIRO: Theoret. Informatics Appl. 25 (1991) 85-100. | Numdam | Zbl 0718.68026

[18] W. Rudin, Real and Complex Analysis, 2nd Ed. Tata McGraw-Hill, New Delhi (1974). | MR 344043 | Zbl 0278.26001

[19] N.R. Scott, Computer Number Systems & Arithmetic. Prentice-Hall, New Jersey (1985). | Zbl 0613.65046

[20] R. Sedgewick and Ph. Flajolet, An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading (1996). | Zbl 0841.68059

[21] I. Wegener, Effiziente Algorithmen für grundlegende Funktionen. B.G. Teubner, Stuttgart (1996). | MR 1076624 | Zbl 0697.68011