An Upper Bound for Sorting $$R_n$$ with LE

• Sai Satwik Kuppili
• 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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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

• Sai Satwik Kuppili
• 1