An Upper Bound for Sorting \(R_n\) with LE

  • Sai Satwik Kuppili
  • Bhadrachalam ChitturiEmail author
  • T. Srinath
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1046)


A permutation on a given alphabet \(\varSigma = (1, 2, 3,\ldots , n)\) is a sequence of elements in the alphabet where every element occurs precisely once. \(S_n\) denotes the set of all such permutations on a given alphabet. \(I_n \in S_n\) be the Identity permutation where elements are in ascending order i.e. \((1, 2, 3,\ldots , n)\). \(R_n \in S_n\) is the reverse permutation where elements are in descending order, i.e. \(R_n =(n, n-1, n-2,\ldots , 2, 1)\). An operation has been defined in OEIS which consists of exactly two moves: set-rotate that we call Rotate and pair-exchange that we call Exchange. Rotate is a left rotate of all elements (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. We call this operation as LE. The optimum number of moves for transforming \(R_n\) into \(I_n\) with LE operation are known for \(n \le 10\); as listed in OEIS with identity A048200. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort \(R_n\) with LE has been derived; (b) the optimum number of moves to sort the next larger \(R_n\) i.e. \(R_{11}\) has been computed. Sorting permutations with various operations has applications in genomics and computer interconnection networks.


Permutations Sorting Cayley graphs Upper bound Set-rotate Pair-exchange 


  1. 1.
    The On-Line Encyclopedia of Integer Sequences.
  2. 2.
    Jerrum, M.R.: The complexity of finding minimum-length generator sequences. Theor. Comput. Sci. 36, 265–289 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Akers, S.B., Krishnamurthy, B.: A group-theoretic model for symmetric interconnection networks. IEEE Trans. Comput. 38(4), 555–566 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chitturi, B., Fahle, W., Meng, Z., Morales, L., Shields, C.O., Sudborough, H.: An (18/11)n upper bound for sorting by prefix reversals. Theor. Comput. Sci. 410(36), 3372–3390 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, T., Skiena, S.S.: Sorting with fixed-length reversals. Discrete Appl. Math. 71(1–3), 269–295 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Feng, X., Chitturi, B., Sudborough, H.: Sorting circular permutations by bounded transpositions. In: Arabnia, H. (ed.) Advances in Computational Biology. AEMB, vol. 680, pp. 725–736. Springer, New York (2010). Scholar
  7. 7.
    Chitturi, B., Sudborough, H., Voit, W., Feng, X.: Adjacent swaps on strings. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 299–308. Springer, Heidelberg (2008). Scholar
  8. 8.
    Lakshmivarahan, S., Jho, J.-S., Dhall, S.K.: Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey. Parallel Comput. 19, 361–407 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chitturi, B.: Perturbed layered graphs. In: ICACCP (2019)Google Scholar
  10. 10.
    Chitturi, B., Balachander, S., Satheesh, S., Puthiyoppil, K.: Layered graphs: applications and algorithms. Algorithms 11(7), 93 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mokhtar, H.: A few families of Cayley graphs and their efficiency as communication networks. Bull. Aust. Math. Soc. 95(3), 518–520 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, T., Gennian, G.: Improved lower bounds on the degree-diameter problem. J. Algebr. Comb. 49, 135–146 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chitturi, B., Das, P.: Sorting permutations with transpositions in \(O(n^3)\) amortized time. Theor. Comput. Sci. 766, 30–37 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Erskine, G., James, T.: Large Cayley graphs of small diameter. Discrete Appl. Math. 250, 202–214 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gostevsky, D.A., Konstantinova, E.V.: Greedy cycles in the star graphs. Discrete Math. Math. Cybern. 15, 205–213 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fertin, G., Labarre, A., Rusu, I., Vialette, S., Tannier, E.: Combinatorics of Genome Rearrangements. MIT Press, Cambridge (2009)CrossRefGoogle Scholar
  17. 17.
    Chitturi, B.: A note on complexity of genetic mutations. Discrete Math. Algorithms Appl. 3(03), 269–286 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Sai Satwik Kuppili
    • 1
  • Bhadrachalam Chitturi
    • 1
    • 2
    Email author
  • T. Srinath
    • 1
  1. 1.Department of Computer Science and EngineeringAmrita Vishwa VidyapeethamAmritapuriIndia
  2. 2.Department of CSUniversity of Texas at DallasRichardsonUSA

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