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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)

Abstract

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.

Keywords

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

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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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