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A003075 Minimal number of comparisons needed for n-element sorting network.
(Formerly M2446)
0, 1, 3, 5, 9, 12, 16, 19, 25, 29, 35, 39 (list; graph; refs; listen; history; text; internal format)



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


R. W. Floyd and D. E. Knuth, The Bose-Nelson sorting problem, pp. 163-172 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, North-Holland, 1973.

H. Jullie, Lecture Notes in Comp. Sci. 929 (1995), 246-260.

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 Nine-Input Sorting Networks", Mathematical Systems Theory, Vol. 24, pp. 101-116, 1991.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..12.

D. Bundala, M. Codish, L. Cruz-Filipe et al., Optimal-Depth Sorting Networks, arXiv preprint arXiv:1412.5302 [cs.DS], 2014.

Michael Codish, Luís Cruz-Filipe, Michael Frank and Peter Schneider-Kamp, Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine 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 (note "A360" refers to N0360 which is A000788).

Jannis Harder, Lower Size Bounds for Sorting Networks

Mariana Nagy, Vlad-Florin Drăgoi and Valeriu Beiu, Employing Sorting Nets for Designing Reliable Computing Nets, IEEE 20th International Conference on Nanotechnology (IEEE-NANO 2020) 370-375.

Ian Parberry, A Computer Assisted Optimal Depth Lower Bound for Nine-Input Sorting Networks

Ed Pegg Jr., Illustration of initial terms

Index entries for sequences related to sorting


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




N. J. A. Sloane


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



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