Abstract
A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r. ISTs can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been widely used to design interconnection networks, constructing ISTs on cayley graphs is worth studying. The alternating group network is a subclass of Cayley graphs, and the approach of constructing ISTs in alternating group networks is still unknown. In this paper, we propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes. Compared to other methods of different interconnection networks in the literature, the uniqueness of our method is that it does not need to determine the parent of one node by any rule, on the contrary others determine that by rules. The time complexity in n-dimensional alternating group network AN n is O(n 2 × n!), where n! is twice the number of nodes of AN n ; hence it is polynomial time. We implement the algorithm in PHP and run cases from AN 3 to AN 10. The results reveal that all spanning trees of AN 3 to AN 10 are independent and that our algorithm is accurate and efficient. INDEX TERMS Independent spanning trees, alternating group networks, triangle breadth-first search, interconnection networks, Cayley graph.
Highlights
A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r, namely that they are internally disjoint paths
In fault-tolerant broadcasting, assume that there exists k ISTs rooted at node r in a network M, and M contains at most k − 1 faulty nodes
The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes
Summary
A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r, namely that they are internally disjoint paths. The IST problem is appealing and has attracted considerable attention It can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution [1], [2]. Each node in the network receives at most one of the k packets except the destination node that receives all the k packets In this regard, the problem of constructing ISTs on graphs is worth studying. The problem is very tough for arbitrary graphs, and researchers have shifted their attention to methods for constructing ISTs in interconnection networks over the past decade. We propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The remainder of this paper is organized as follows: Section II explains preliminary considerations; Section III presents the algorithm; Section IV proves the relevant claims; Section V describes the software implementation; and Section VI concludes the paper
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