Transposition Tree Research Articles

Token Swapping on Trees

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.

Optimal Algorithms for Sorting Permutations with Brooms

Sorting permutations with various operations has applications in genetics and computer interconnection networks where an operation is specified by its generator set. A transposition tree T=(V,E) is a spanning tree over n vertices v1,v2,…vn. T denotes an operation in which each edge is a generator. A value assigned to a vertex is called a token or a marker. The markers on vertices u and v can be swapped only if the pair (u,v)∈E. The initial configuration consists of a bijection from the set of vertices v1,v2,…,vn to the set of markers (1,2,⋯,n−1,n). The goal is to sort the initial configuration of T, i.e., an input permutation, by applying the minimum number of swaps or moves in T. Computationally tractable optimal algorithms to sort permutations are known only for a few classes of transposition trees. We study a class of transposition trees called a broom and its variation a double broom. A single broom is a tree obtained by joining the centre vertex of a star with one of the two leaf vertices of a path graph. A double broom is an extension of a single broom where the centre vertex of a second star is connected to the terminal vertex of the path in a single broom. We propose a simple and efficient algorithm to obtain an optimal swap sequence to sort permutations with the transposition tree broom and a novel optimal algorithm to sort permutations with a double broom. We also introduce a new class of trees named millipede tree and prove that D* yields a tighter upper bound for sorting permutations with a balanced millipede tree compared to D′. Algorithms D* and D′ are designed previously.

Fault-Tolerant Strongly Hamiltonian Laceability and Hyper-Hamiltonian Laceability of Cayley Graphs Generated by Transposition Trees

AbstractA bipartite graph is Hamiltonian laceable if any two of its vertices in different partite sets are connected by a Hamiltonian path. A Hamiltonian laceable graph $G$ is called strongly Hamiltonian laceable if any two of its vertices in the same partite set are connected by a path of length $|V(G)|-2$. A Hamiltonian laceable graph $G$ (with two partite sets $V_0, V_1$) is called hyper-Hamiltonian laceable, if for any vertex $v \in V_{i}$ for $i \in \{0,1\}$, there is a Hamiltonian path of $G-\{v\}$ between any two vertices in $V_{1-i}$. In this paper, we focus on the edge-fault-tolerant strongly Hamiltonian laceability and hyper-Hamiltonian laceability on the class of Cayley graphs generated by transposition trees, which are a generalization of star graph and bubble-sort graph. For every $n$-dimensional Cayley graph generated by a transposition tree $\Gamma _n$, we show that $\Gamma _{n}-F$ is strongly Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-3$, which generalizes results in [ 1, 11], and show that $\Gamma _{n}-F$ is hyper-Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-4$.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.

The preclusion numbers and edge preclusion numbers in a class of Cayley graphs

Let G be a hierarchical network (graph) with vertex set V(G) and edge set E(G). The preclusion set of the subnetwork G′ (defined as a smaller network but with the same topological properties as the original one) in G is a subset V′ of V(G) such that G−V′ has no any subnetwork isomorphic to G′. The preclusion number of G′ in G is F(G′)=min{|V′|:V′ is the preclusion set of G′}. Similarly, the edge preclusion set of G′ in G is a subset E′ of E(G) such that G−E′ has no any subnetwork isomorphic to G′. The edge preclusion number of G′ in G is f(G′)=min{|E′|:E′ is the edge preclusion set of G′}. The preclusion number and edge preclusion number are parameters which measure the fault tolerance of interconnection networks in the presence of failures. In this paper, we investigate the two parameters for a class of Cayley graphs T2n+1 generated by the transposition tree similar to the star. Let m be an integer with 0≤m≤n−1, let F(n,m)=F(T2(n−m)+1), and let f(n,m)=f(T2(n−m)+1). We prove that F(n,0)=f(n,0)=1, F(n,1)=2n(2n+1) with 2n+1 prime, f(n,1)≤4n(2n+1) with 2n+1 prime, F(n,n−1)=(2n+1)!6, f(n,n−1)=n(2n+1)!6, and 2n+12m(2m)!≤F(n,m)≤f(n,m)≤nm2n+12m(2m)!. Finally, we prove that F(n,m) is of order at most n3m−1.

An Upper Bound For Sorting Permutations With A Transposition Tree

Abstract Cayley graphs are constructed over n! vertices that denote n! permutations of the symmetric group Sn. The edge set of a particular Cayley graph Γ is determined by the corresponding operation that consists of a well defined generator set. One such operation is a transposition tree which is a spanning tree over n vertices. Cayley graphs have been extensively studied and they have applications in genomic studies and computer interconnection networks. In computer interconnection networks each vertex of the corresponding Γ denotes a computer and an existence of an edge between two vertices indicates that a direct two way communication exists between the corresponding computers. In 2013, upper bounds for Cayley graphs generated by transposition trees were computed in polynomial time. One such upper bound δ´ was known to be the tightest until recently. In 2019, δ* was shown to be better than δ’ in a cumulative sense. We improve upon δ’* to obtain δ’* i is better than δ’ in a cumulative sense, as indicated by the experimental results. Likewise, δ’* is comparable to δ*. For corresponding classes of trees, δ’* is shown to be tighter than δ’ and δ* respectively.

Nonexistence of efficient dominating sets in the Cayley graphs generated by transposition trees of diameter 3

Let d,n be positive integers such that d<n, and let Xnd be a Cayley graph generated by a transposition tree of diameter d. It is known that every Xnd with d<3 splits into efficient dominating sets. The main result of this paper is that Xn3 does not have efficient dominating sets.

The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM⁎ model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. In 2012, Peng et al. proposed a new measure for fault diagnosis of the system, which is called g-good-neighbor diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a favorable topology structure of interconnection networks, the Cayley graph CΓn generated by the transposition tree Γn has many good properties. In this paper, we give that the 2-good-neighbor diagnosability of CΓn under the PMC model and MM⁎ model is g(n−2)−1, where n≥4 and g is the girth of CΓn.

UPPER BOUNDS FOR SORTING PERMUTATIONS WITH A TRANSPOSITION TREE

An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g., interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n2) time method that examines n! permutations to compute an upper bound, f(Γ), to sort any permutation with a given transposition tree T, where Γ is the Cayley graph corresponding to T. We compute two intuitive upper bounds γ and δ′ each in O(n2) time for the same, by working solely with the transposition tree. Recently, Ganesan computed β, an estimate of the exact upper bound for the same, in O(n2) time. Our upper bounds are tighter than f(Γ) and β, on average and in most of the cases. For a class of trees, we prove that the new upper bounds are tighter than β and f(Γ).

FAULT-TOLERANT MAXIMAL LOCAL-CONNECTIVITY ON CAYLEY GRAPHS GENERATED BY TRANSPOSITION TREES

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. In this paper, we define two vertices as maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. Moreover, we show that an (n-1)-regular Cayley graph generated by transposition tree is maximally local-connected, even if there are at most (n-3) faulty vertices in it, and prove that it is also (n-1)-fault-tolerant one-to-many maximally local-connected.

Linearly many faults in Cayley graphs generated by transposition trees

We prove that when linearly many vertices are deleted in a Cayley graph generated by a transposition tree, the resulting graph has a large connected component containing almost all remaining vertices.

Routing Function and Deadlock Avoidance in a Star Graph Interconnection Network

In this paper the properties of paths in a star graph are investigated through the analysis of the corresponding star transposition tree. The general algebraic expression for all shortest paths between any two nodes (routing function) is found, and it is shown that every shortest path consists of a number of subpaths which can be combined in an arbitrary order or even mutually nested. Further, due to the known routing function the deadlock problem is solved using the method of virtual channels. A minimal deterministic routing algorithm is developed which recognizes the structure of the path by extracting subpaths and allows optimal adaptive management of virtual channels. Finally, based on the sufficient number of virtual channels, the minimal fully adaptive routing algorithm is suggested which offers an opportunity to reroute the message a number of times, while maintaining the shortest path between two nodes.