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

Abstract

1 Abstract-A permutation π over alphabet Σ = 1, 2, 3, . . . , n, is a sequence where every element x in Σ occurs exactly once. Sn is the symmetric group consisting of all permutations of length n defined over Σ. In = (1, 2, 3, . . . , n) and Rn = (n, n − 1, n − 2, . . . , 2, 1) are identity (i.e. sorted) and reverse permutations respectively. An operation, that we call as an LRE operation, has been defined in OEIS with identity A186752. This operation is constituted by three generators: left-rotation, rightrotation and transposition(1,2). We call transposition(1,2) that swaps the two leftmost elements as Exchange. The minimum number of moves required to transform Rn into In with LRE operation are known for n ≤ 11 as listed in OEIS with sequence number A186752. For this problem no upper bound is known. OEIS sequence A186783 gives the conjectured diameter of the symmetric group Sn when generated by LRE operations [1]. The contributions of this article are: (a) The first non-trivial upper bound for the number of moves required to sort Rn with LRE; (b) a tighter upper bound for the number of moves required to sort Rn with LRE; and (c) the minimum number of moves required to sort R10 and R11 have been computed. Here we are computing an upper bound of the diameter of Cayley graph generated by LRE operation. Cayley graphs are employed in computer interconnection networks to model efficient parallel architectures. The diameter of the network corresponds to the maximum delay in the network.