Sorting permutations with a transposition tree
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The set of all permutations with n symbols is a symmetric group denoted by S_n. A transposition tree, T, is a spanning tree over its n vertices V_T=1, 2, 3, ... n where the vertices are the positions of a permutation π and π is in S_n. T is the operation and the edge set E_T denotes the corresponding generator set. The goal is to sort a given permutation π with T. The number of generators of E_T that suffices to sort any π∈ S_n constitutes an upper bound. It is an upper bound, on the diameter of the corresponding Cayley graph Γ i.e. diam(Γ). A precise upper bound equals diam(Γ). Such bounds are known only for a few trees. Jerrum showed that computing diam(Γ) is intractable in general if the number of generators is two or more whereas T has n-1 generators. For several operations computing a tight upper bound is of theoretical interest. Such bounds have applications in evolutionary biology to compute the evolutionary relatedness of species and parallel/distributed computing for latency estimation. The earliest algorithm computed an upper bound f(Γ) in a Ω(n!) time by examining all π in S_n. Subsequently, polynomial time algorithms were designed to compute upper bounds or their estimates. We design an upper bound δ^* whose cumulative value for all trees of a given size n is shown to be the tightest for n ≤ 15. We show that δ^* is tightest known upper bound for full binary trees. Keywords: Transposition trees, Cayley graphs, permutations, sorting, upper bound, diameter, greedy algorithms.
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