Independent Spanning Trees Research Articles

Independent Spanning Trees on Multidimensional Torus Networks

Two spanning trees rooted at vertex r in a graph G are called independent spanning trees (ISTs) if for each vertex v in G, vner, the paths from vertex v to vertex r in these two trees are internally distinct. If the connectivity of G is k, the IST problem is to construct k ISTs rooted at each vertex. The IST problem has found applications in fault-tolerant broadcasting, but it is still open for general graphs with connectivity greater than four. In this paper, we shall propose a very simple algorithm for solving the IST problem on multidimensional torus networks. In our algorithm, every vertex can determine its parent for a specific independent spanning tree only depending on its own label. Thus, our algorithm can also be implemented in parallel systems or distributed systems very easily.

Optimal Independent Spanning Trees on Odd Graphs

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. The designs of multiple ISTs on several classes of networks have been widely investigated. In this paper we show a construction algorithm of ISTs on odd graphs, and we analyze that all the lengths of the paths in the ISTs are less than or equal to the length of the shortest path+4, which is optimal. We also prove that the heights of the ISTs we constructed are d+1, which again is optimal, since the fault diameter of an odd graph is d+1.

On the independent spanning trees of recursive circulant graphs [formula omitted] with [formula omitted

Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r , and for each vertex v ≠ r in G , the two different paths from v to r , one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. The construction of multiple independent spanning trees has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. A recursive circulant graph G ( N , d ) has N = c d m vertices labeled from 0 to N − 1 , where d ⩾ 2 , m ⩾ 1 , and 1 ⩽ c < d , and two vertices x , y ∈ G ( N , d ) are adjacent if and only if there is an integer k with 0 ⩽ k ⩽ ⌈ log d N ⌉ − 1 such that x ± d k ≡ y (mod N ). In this paper, we propose an algorithm to construct multiple independent spanning trees on a recursive circulant graph G ( c d m , d ) with d > 2 , where the number of independent spanning trees matches the connectivity of G ( c d m , d ) .

Reducing the Height of Independent Spanning Trees in Chordal Rings

This paper is concerned with a particular family of regular 4-connected graphs, called chordal rings. Chordal rings are a variation of ring networks. By adding two extra links (or chords) at each vertex in a ring network, the reliability and fault-tolerance of the network are enhanced. Two spanning trees on a graph are said to be independent if they are rooted at the same vertex, say, r, and for each vertex v \neq r, the two paths from r to v, one path in each tree, are internally disjoint. A set of spanning trees on a given graph is said to be independent if they are pairwise independent. Iwasaki et al. [CHECK END OF SENTENCE] proposed a linear time algorithm for finding four independent spanning trees on a chordal ring. In this paper, we give a new linear time algorithm to generate four independent spanning trees with a reduced height in each tree. Moreover, a complete analysis of our improvements on the heights of independent spanning trees is also provided.

Parallel construction of optimal independent spanning trees on hypercubes

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. Tang et al. [S.-M. Tang, Y.-L. Wang, Y.-H. Leu, Optimal independent spanning trees on hypercubes, Journal of Information Science and Engineering 20 (2004) 143–155] studied the problem of constructing k ISTs on k-dimensional hypercube Q k , and provided a recursive algorithm for their construction (i.e., for constructing k ISTs of Q k , it needs to build k − 1 ISTs of Q k−1 in advance). This kind of construction forbids the possibility that the algorithm could be parallelized. In this paper, based on a simple concept called Hamming distance Latin square, we design a new algorithm for generating k ISTs of Q k . The newly proposed algorithm relies on a simple rule and is easy to be parallelized. As a result, we show that the ISTs we constructed are optimal in the sense that both the heights and the average path length of trees are minimized.

Independent spanning trees with small depths in iterated line digraphs

We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths.

Independent spanning trees of chordal rings

A chordal ring, denoted by CR( N, d), is a graph G = ( V, E) with V = {0, 1,…, N − 1} and E = {(u, v) ¦[v − u] N = 1 or d} , where 2 ⩽ N ⩽ N 2 and [ r] N denotes r modulo N. We show that for 2 ⩽ d < N 2 , CR(N, d) has four independent spanning trees rooted at the same vertex.