Sorting permutations with a transposition tree

Abstract

The set of all permutations with n symbols is a symmetric group denoted by Sn. A transposition tree, T , is a spanning tree over its n vertices VT =1, 2, 3, . . . n where the vertices are the positions of a permutation π and π is in Sn. T is the operation and the edge set ET denotes the corresponding generator set. The goal is to sort a given permutation π with T . The number of generators of ET that suffices to sort any π ∈ Sn 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 tress. 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 Sn. 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. 4

Optimal Algorithms for Sorting Permutations with Brooms

“…permutations. Subsequently, various polynomial time approximation algorithms were designed [23][24][25]29,30]. Chitturi introduced algorithm D in [24] which finds the upper bound δ in polynomial time, which is the tightest known upper bound when the upper bounds of all trees are summed together.…”

“…Chitturi introduced algorithm D in [24] which finds the upper bound δ in polynomial time, which is the tightest known upper bound when the upper bounds of all trees are summed together. Subsequently, algorithm D * which identifies corresponding the upper bound δ * was designed in [29]. It was shown that δ * is tighter for balanced-starburst tree [29].…”

“…Subsequently, algorithm D * which identifies corresponding the upper bound δ * was designed in [29]. It was shown that δ * is tighter for balanced-starburst tree [29]. In [31], an analysis of both D and D * was performed and shows that δ * is tighter for full binary tree too.…”

Hardness of Token Swapping on Trees

“…Sequential token swapping has been studied before by many researchers in many disparate fields, from discrete mathematics [8,34,37,33,38,29] and theoretical computer science [25,28,19,40,42,3,32,7,26,41,11,10,6] to more applied fields including network engineering as mentioned earlier [1], robot motion planning [14,36], and game theory [22]. On general graphs, the problem is known to be NP-complete [3], furthermore APX-hard [32], and even W [1]-hard with respect to number of swaps [7].…”

“…They gave an algorithm for finding short (but not necessarily shortest) paths in the resulting Cayley graphs, and characterized the diameter of the Cayley graph (and thus found optimal paths in the worst case over possible start/destination pairs of vertices) when the tree is a star. Follow-up work along this line attains tighter upper bounds on the diameter of the Cayley graph in this situation when the graph is a tree [37,19,29,11] and develops exponential algorithms to compute the exact diameter of the Cayley graph of a transposition tree [10], though the complexity of the latter problem remains open. Here we show NP-hardness of the slightly more general problem of computing the shortest-path distance between two given nodes in the Cayley graph of a transposition tree (sequential token swapping on a tree), a problem implicit in [1] and explicitly posed as an open problem in [40].…”