Elsevier

Information Sciences

Volume 240, 10 August 2013, Pages 191-204
Information Sciences

The number of shortest paths in the arrangement graph

https://doi.org/10.1016/j.ins.2013.03.035Get rights and content

Abstract

A solution to the shortest path enumeration problem can find numerous applications in studying issues related to interconnection networks. In this paper, we enumerate the number of shortest paths between any two vertices in an arrangement graph by establishing a bijection between these shortest paths and a collection of ordered forests of certain bi-colored trees, via minimum factorizations of permutations corresponding to such vertices in terms of so-called arrangement transpositions, and then count the number of these forests with the help of an existing result on the enumeration of such bi-colored trees.

Our result generalizes previous ones and can be applied to solve the same problem for other related graphs such as the alternating group graph. On the other hand, the techniques applied to derive such an enumeration result further extend the ongoing work of counting the number of minimum factorizations of a permutation in terms of a certain type of transposition, a rather interesting problem in the area of algebraic combinatorics.

Introduction

Given a graph G, a basic and interesting problem is to enumerate the number of shortest paths, not necessarily disjoint, between a pair of vertices in G. A solution to this problem can provide an important topological property of an interconnection network, in terms of its connectivity, fault-tolerance, communication expense, and routing flexibility [18]. For the history of this problem and its connection to topology, readers are referred to [10], [12] and the references cited within.

This shortest path enumeration problem is not trivial in general. On the other hand, it is easy to see that there are H(u  v)! shortest paths between u and v, two vertices in a hypercube, where H(·) is the Hamming weight function, and ⊕ the exclusive-or operation. An explicit formula for this quantity has been obtained for the hexagonal network [8] and the star graph [13]. Such a result has also been achieved recently for the (n, k)-star graph [2], [3] following two different approaches, both based on a combinatorial result established in [13]. This latter result, establishing a bijection between the shortest paths in the star graph and a collection of bi-colored trees and then counting the bi-colored trees, is interesting in its own right.

In this paper, we enumerate the number of shortest paths between any two vertices in an arrangement graph by reducing this problem to counting the number of minimum factorizations of a permutation into a certain type of transposition. Starting with Hurwitz’s work dating back to the 19th century [11], this general counting problem has been of great and refreshing interest to both theoreticians and practitioners. Dénes counted this number in [6] for an arbitrary permutation where no restriction is placed on the transposition type. Stanley studied the same problem, where the allowed transpositions form a Coxeter group [19], i.e., those in the form of (i, i + 1), i  [1, n), corresponding to edges in a bubblesort graph [15], where “i  [1, n)” stands for “1  i < n”; and Irving, while further extending a combinatorial result achieved by Pak [17], solved this problem for the star transpositions [13], (1, i), i  [2, n], corresponding to edges in a star graph [1]. (Irving actually counted the number of minimum and transitive factorizations of a permutation in terms of the star transpositions. For more recent work on minimum and transitive factorizations of a permutation in terms of the star transpositions and its relation to the Jucys-Murphy elements, readers are referred to [9], [16], [20], [7] and references cited therein.) Recently, the number of minimum factorizations of a permutation in terms of the extended star transpositions was also derived in [3], which agrees with the result achieved in [2] in giving the number of shortest paths between any two vertices in the (n, k)-star graphs. We now further investigate this problem by counting the number of minimum factorizations of a permutation in terms of the arrangement transpositions, in the form of (e, i), e  (k, n] and i  [1, k], where k  [1, n) and n  3. It turns out that this latter type of transposition is associated with the arrangement graph, introduced in [5].

The rest of this paper is organized as follows. After introducing in the next section basic notions and results related to the arrangement graph, in particular, the notion of a minimum factorization of a vertex in such a graph, we discuss the structure of such a minimum factorization and the relationship between such minimum factorizations and shortest paths in Section 3. We then establish a bijection between these minimum factorizations and a set of ordered forests in Section 4, and enumerate the number of such ordered forests in Section 5. We finally conclude this paper in Section 6.

Section snippets

Minimum factorizations of vertices in an arrangement graph

The arrangement graph preserves many nice properties of the popular star graph [1], such as having small (average) diameter, a hierarchical structure, vertex and edge symmetry, simple shortest path routing, and various fault tolerance properties, while bringing a solution to the scalability issue associated with the star graph: A star graph with n dimensions, denoted by Sn, has n! vertices, while an arrangement graph with n dimensions, and an additional parameter k  [1, n), has n!/(n  k)!

Structure of minimum factorizations

To enumerate all the minimum factorizations, we need to explore their structures in terms of the factors. Their structures in terms of the star transpositions, which swap elements located in position i (∈ [2, n]) and position 1, have been characterized in[13]. We now discuss the case in terms of the arrangement transpositions, associated with the arrangement graph An,k, which swap elements between some internal position i  [1, k] and some external position e  (k, n]. A key observation here is that Sn

Relating minimum factorizations to a forest

A bijection is established in Irving[13] from the transitive minimum factorizations of a vertex in a star graph to a collection of bi-colored trees. We first generalize Irving’s bijection result by establishing a relationship from the minimum factorizations of a vertex in An,k to a collection of ordered forests of bi-colored trees.

Let f(π) be a minimum factorization of π  V(An,k). We associate a term, 〈κ, C, e〉, with every factor of f(π), where κ refers to the index of the cycle that this factor

The number of shortest paths in an arrangement graph

Given a vertex πV(An,k),πEk, the factorization of an external cycle of C(π) is clearly fixed. On the other hand, the factorization of an internal cycle of C(π) depends on which external symbol is assigned to such a cycle, and whether it shares this symbol with an external cycle or other internal cycles. Thus to enumerate the number of minimum factorizations of a vertex, we need to specify such an assignment of external symbols to internal cycles, which can be done as follows. We partition all

Concluding remarks

In this paper, we further extended the ongoing work of counting the number of minimum factorizations of a permutation in terms of a certain type of transposition. In particular, we enumerated the number of minimum factorizations of permutations on 〈n〉, n  3, in terms of arrangement transpositions, i.e., transpositions of the form (e, i) where e  (k, n] and i  [1, k], k  [1, n  1]. On the practical side, this result immediately leads to the number of the shortest paths between any two vertices in the

Acknowledgments

We greatly appreciate the thorough review and insightful comments made by an anonymous referee, which led to a clarification and further improvement of this paper.

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