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A Note on a Problem Posed by D. E. Knuth on a Satisfiability Recurrence

Published online by Cambridge University Press:  09 July 2014

PHILIPPE JACQUET ,
CHARLES KNESSL  and
WOJCIECH SZPANKOWSKI
PHILIPPE JACQUET
Affiliation:
Bell Labs, Alcatel-Lucent, 91620 Nozay, France (e-mail: philippe.jacquet@inria.fr)
CHARLES KNESSL
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA (e-mail: knessl@uic.edu)
WOJCIECH SZPANKOWSKI
Affiliation:
Department of Computer Science, Purdue University, W. Lafayette, IN 47907, USA (e-mail: spa@cs.purdue.edu)
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Abstract

We resolve a conjecture proposed by D.E. Knuth concerning a recurrence arising in the satisfiability problem. Knuth's recurrence resembles recurrences arising in the analysis of tries, in particular PATRICIA tries, and asymmetric leader election. We solve Knuth's recurrence exactly and asymptotically, using analytic techniques such as the Mellin transform and analytic depoissonization.

Keywords

Primary 68W40
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 829 - 841
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Davis, M., Logemann, G. and Loveland, D. (1962) A machine program for theorem proving. Comm. Assoc. Comput. Mach. 5 394–397.Google Scholar
[2]Fayolle, G., Flajolet, P. and Hofri, M. (1986) On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel. Adv. Appl. Probab. 18 441–472.Google Scholar
[3]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.Google Scholar
[4]Flajolet, P., Gourdon, X. and Dumas, P. (1995) Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 3–58.Google Scholar
[5]Goldberg, A. (1979) On the complexity of the satisfiability problem. Courant Computer Science Report no. 16, New York University.Google Scholar
[6]Goldberg, A., Purdom, P. and Brown, C. (1982) Average time analyses of simplified Davis–Putnam procedures. Inform. Process. Lett. 15 72–75.CrossRefGoogle Scholar
[7]Jacquet, P. and Szpankowski, W. (1998) Analytical depoissonization and its applications. Theoret. Comput. Sci. 201 1–62.Google Scholar
[8]Janson, S. and Szpankowski, W. (1997) Analysis of an asymmetric leader election algorithm. Electron. J. Combin. 4 R17.Google Scholar
[9]Knuth, D. E. (1998) The Art of Computer Programming: Sorting and Searching, Vol. 3, second edition, Addison-Wesley.Google Scholar
[10]Knuth, D. E. personal communication.Google Scholar
[11]Rais, B., Jacquet, P. and Szpankowski, W. (1993) Limiting distribution for the depth in Patricia tries. SIAM J. Discrete Math. 6 197–213.Google Scholar
[12]Sedgewick, R. personal communication.Google Scholar
[13]Szpankowski, W. (1990) Patricia tries again revisited. J. Assoc. Comput. Mach. 37 691–711.Google Scholar
[14]Szpankowski, W. (2001) Analysis of Algorithms on Sequences, Wiley.Google Scholar