

A003075


Minimal number of comparisons needed for nelement sorting network.
(Formerly M2446)


6



0, 1, 3, 5, 9, 12, 16, 19, 25, 29, 35, 39
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OFFSET

1,3


COMMENTS

It is conjectured that the sequence continues (after 39) 45, 51, 56, 60, ...
a(13) <= 45 is mentioned in Knuth, Sorting and Searching, Vol. 2. a(9) was determined in 1991.  Ed Pegg Jr, Dec 05 2001.
Correction: the value for a(9) was not determined in the 1991 reference, which instead is about optimal depth.  Michael Codish, Jun 01 2014


REFERENCES

R. W. Floyd and D. E. Knuth, The BoseNelson sorting problem, pp. 163172 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, NorthHolland, 1973.
H. Jullie, Lecture Notes in Comp. Sci. 929 (1995), 246260.
D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.4, Eq. (11).
I. Parberry, "A Computer Assisted Optimal Depth Lower Bound for NineInput Sorting Networks", Mathematical Systems Theory, Vol. 24, pp. 101116, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..12.
D. Bundala, M. Codish, L. CruzFilipe et al., OptimalDepth Sorting Networks, arXiv preprint arXiv:1412.5302 [cs.DS], 2014.
Michael Codish, Luís CruzFilipe, Michael Frank, Peter SchneiderKamp, TwentyFive Comparators is Optimal when Sorting Nine Inputs (and TwentyNine for Ten), arXiv:1405.5754 [cs.DM], 2014.
Bert Dobbelaere, Smallest and fastest sorting networks for a given number of inputs
Milton W. Green, Letter to N. J. A. Sloane, 1973.
Jannis Harder, Lower Size Bounds for Sorting Networks
Mariana Nagy, VladFlorin Drăgoi, Valeriu Beiu, Employing Sorting Nets for Designing Reliable Computing Nets, IEEE 20th International Conference on Nanotechnology (IEEENANO 2020) 370375.
Ian Parberry, A Computer Assisted Optimal Depth Lower Bound for NineInput Sorting Networks
Ed Pegg Jr., Illustration of initial terms
Index entries for sequences related to sorting


CROSSREFS

A006282 is an upper bound. Cf. A036604, A067782 (minimal depth).
Sequence in context: A191403 A233779 A287105 * A061562 A006282 A086845
Adjacent sequences: A003072 A003073 A003074 * A003076 A003077 A003078


KEYWORD

hard,nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Updates from Ed Pegg Jr, Dec 05 2001
Correction and update: terms are exact for n<=10. The precise values for n=9 and n=10 are established in the reference from 2014 by Codish et al.  Michael Codish, Jun 01 2014
Entry revised by N. J. A. Sloane, Jun 02 2014
a(11)a(12) from Jannis Harder, Dec 10 2019


STATUS

approved



