An Upper Bound for Sorting R_n with LRE
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A permutation π over alphabet Σ = 1,2,3,...,n, is a sequence where every element x in Σ occurs exactly once. S_n is the symmetric group consisting of all permutations of length n defined over Σ. I_n = (1, 2, 3,..., n) and R_n =(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, right-rotation 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 R_n into I_n 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 S_n when generated by LRE operations <cit.>. The contributions of this article are: (a) The first non-trivial upper bound for the number of moves required to sort R_n with LRE; (b) a tighter upper bound for the number of moves required to sort R_n with LRE; and (c) the minimum number of moves required to sort R_10 and R_11 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.
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