Abstract
A (σ,k)-de Bruijn sequence is a minimum length string on an alphabet set of size σ which contains all σ k k-mers exactly once. Motivated by an application in synthetic biology, we say a given collection of de Bruijn sequences are orthogonal if no two of them contain the same (k + 1)-mer; that is, the length of their longest common substring is k.
In this paper, we show how to construct large collections of orthogonal de Bruijn sequences. In particular, we prove that there are at least \(\lfloor \sigma/2 \rfloor\) mutually-orthogonal order-k de Bruijn sequences on alphabets of size σ for all k. Based on this approach, we present a heuristic which proves capable of efficiently constructing optimal collections of mutually-orthogonal sequences for small values of σ and k, which supports our conjecture that σ − 1 mutually-orthogonal de Bruijn sequences exist for all σ and k.
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References
Bermond, J.-C.: Hamiltonian decompositions of graphs, directed graphs and hypergraphs. Ann. Discrete Math. 3, 21–28 (1978); Présentéau Cambridge Combinatorial Conf., Advances in Graph Theory , Trinity College, Cambridge, England (1977)
Bermond, J.-C., Darrot, E., Delmas, O., Perennes, S.: Hamilton circuits in the directed wrapped butterfly network. Discrete Applied Mathematics 84(1), 21–42 (1998)
Bond, J., Iványi, A.: Modelling of interconnection networks using de bruijn graphs. In: Iványi, A. (ed.) Third Conference of Program Designer, Budapest (1987)
Bugl, H., Danner, J.P., Molinari, R.J., Mulligan, J.T., Park, H.-O., Reichert, B., Roth, D.A., Wagner, R., Budowle, B., Scripp, R.M., Smith, J.A.L., Steele, S.J., Church, G., Endy, D.: DNA synthesis and biological security. Nature Biotechnology 25, 627–629 (2007)
Coleman, J.R., Papamichial, D., Futcher, B., Skiena, S., Mueller, S., Wimmer, E.: Virus attenuation by genome-scale changes in codon-pair bias. Science 320, 1784–1787 (2008)
Czar, M.J., Anderson, J.C., Bader, J.S., Peccoud, J.: Gene synthesis demystified. Trends in Biotechnology 27(2), 63–72 (2009)
de Bruijn, N.G.: A combinatorial problem. Koninklijke Nederlandse Akademie v. Wetenschappen 49, 758–764 (1946)
Gibson, D., et al.: Creation of a bacterial cell controlled by a chemically synthesized genome. Science (2010), doi:10.1125./science.1190719
Fleischner, H., Jackson, B.: Compatible euler tours in eulerian digraphs. In: Cycles and Rays, Proceeding Colloquium Montreal, 1987. ATO ASI Ser. C, pp. 95–100. Kluwer Academic Publishers, Dordrecht (1990)
Golomb, S.W.: Shift Register Sequences. Holden-Day (1967)
Good, I.J.: Normal recurring decimals. J. London Math. Soc. 21, 167–172 (1946)
Kása, Z.: On arc-disjoint hamiltonian cycles in de Bruijn graphs. CoRR abs/1003.1520 (2010)
Kautz, W.H.: Bounds on directed (d,k) graphs. In: Theory of Cellular Logic Networks and Machines, AFCKL-68-0668 Final Rep., vol. 24, pp. 20–28 (1968)
Knuth, D.E.: Oriented subtrees of an arc digraph. Journal of Combinatorial Theory 3, 309–314 (1967)
Montes, P., Memelli, H., Ward, C., Kim, J., Mitchell, J., Skiena, S.: Optimizing restriction site placement for synthetic genomes. In: Amir, A., Parida, L. (eds.) CPM 2010. LNCS, vol. 6129, pp. 323–337. Springer, Heidelberg (2010)
Mueller, S., Coleman, R., Papamichail, D., Ward, C., Nimnual, A., Futcher, B., Skiena, S., Wimmer, E.: Live attenuated influenza vaccines by computer-aided rational design. Nature Biotechnology 28 (2010)
Ronse, C.: Feedback Shift Registers. Springer, Berlin (1984)
Rosenfeld, V.R.: Enumerating Kautz sequences. Kragujevac Journal of Mathematics 24, 19–41 (2002)
Rowley, R., Bose, B.: Edge-disjoint Hamiltonian cycles in de Bruijn networks. In: Distributed Memory Computing Conference, pp. 707–709 (1991)
Rowley, R., Bose, B.: On the number of arc-disjoint Hamiltonian circuits in the de Bruijn graph. Parallel Processing Letters 3(4), 375–380 (1993)
Tutte, W.T.: The dissection of equilateral triangles into equilateral triangles. Mathematical Proceedings of the Cambridge Philosophical Society 44, 463–482 (1948)
van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Stevin: Wisen Natuurkundig Tijdschrift 28, 203–217 (1951)
West, D.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (2000)
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Lin, YL., Ward, C., Jain, B., Skiena, S. (2011). Constructing Orthogonal de Bruijn Sequences. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_50
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DOI: https://doi.org/10.1007/978-3-642-22300-6_50
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