Multiple Independent Spanning Trees Research Articles

A parallel algorithm for constructing multiple independent spanning trees in bubble-sort networks

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 secure message distribution. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. Kao et al. (2019) [18] proposed an algorithm to construct independent spanning trees in bubble-sort networks. The algorithm is executed in a recursive function and thus is hard to parallelize. In this paper, we focus on the problem of constructing ISTs in bubble-sort networks Bn and present a non-recursive algorithm. Our approach can be fully parallelized, i.e., every vertex can determine its parent in each spanning tree in constant time. This solves the open problem from the paper by Kao et al. Furthermore, we show that the total time complexity O(n⋅n!) of our algorithm is asymptotically optimal, where n is the dimension of Bn and n! is the number of vertices of the network.

Parallel construction of multiple independent spanning trees on highly scalable datacenter networks

An emerging datacenter network (DCN) with high scalability called HSDC is a server-centric DCN that can help cloud computing in supporting many inherent cloud services. For example, a server-centric DCN can initiate routing for data transmission. This paper investigates the construction of independent spanning trees (ISTs for short), a set of the rooted spanning trees associated with the disjoint-path property, in HSDC. Regarding multiple spanning trees as routing protocol, ISTs have applications in data transmission, e.g., fault-tolerant broadcasting and secure message distribution. We first establish the vertex-symmetry of HSDC. Then, by the structure that n-dimensional HSDC is a compound graph of an n-dimensional hypercube Qn and n-clique Kn, we amend the algorithm constructing ISTs for Qn to obtain the algorithm required by HSDC. Unlike most algorithms of recursively constructing tree structures, our algorithm can find every node’s parent in each spanning tree directly via an easy computation relied upon only the node address and tree index. Consequently, we can implement the algorithm for constructing n ISTs in O(nN) time, where N=n2n is the number of vertices of n-dimensional HSDC; or parallelize the algorithm in O(n) time using N processors. Remarkably, the diameter of the constructed ISTs is about twice the diameter of Qn.

The Construction of Multiple Independent Spanning Trees on Burnt Pancake Networks

A set of the spanning trees in a graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called independent spanning trees if they have a common root <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> and for each vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v\in V(G)\setminus \{r\}$ </tex-math></inline-formula> , the paths from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> in any two trees are directed edge-disjoint and internally vertex-disjoint. The construction of independent spanning trees has many practical applications in reliable communication networks, such as fault-tolerant transmission and secure message distribution. A burnt pancake network <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$BP_{n}$ </tex-math></inline-formula> is a kind of Cayley graph, which has been proposed as the topology of an interconnection network. In this paper, we provide a two stages construction scheme that can be used to construct a maximal number of independent spanning trees on a burnt pancake network in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N\times n)$ </tex-math></inline-formula> time, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is the number of nodes of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$BP_{n}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the dimension of the network. Furthermore, we prove the correctness of our proposed algorithm in constructing independent spanning trees.

Amortized efficiency of constructing multiple independent spanning trees on bubble-sort networks

A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex $$v(\ne r)$$ in G, the two paths from v to r in any two trees share no common edge and no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design the topologies of interconnection networks, construction of ISTs on Cayley graphs is significative. It is well-known that star networks $$S_n$$ and bubble-sort network $$B_n$$ are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on $$S_n$$ (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on $$B_n$$ is not dealt with yet. In this paper, we present an algorithm to construct $$n-1$$ ISTs of $$B_n$$ . Moreover, we show that our algorithm has amortized efficiency for multiple trees construction. In particular, every vertex can determine its parent in each spanning tree in a constant amortized time. Accordingly, except for the star networks, it seems that our work is the latest breakthrough on the problem of ISTs for all subfamilies of Cayley graphs.

Parallel Construction of Independent Spanning Trees on Enhanced 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, bandwidth and security. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. In this paper, we give an algorithm to construct ISTs on enhanced hypercubes $Q_{n,k}$ , which contain folded hypercubes as a subclass. Moreover, we show that these ISTs are near optimal for heights and path lengths. Let $D(Q_{n,k})$ denote the diameter of $Q_{n,k}$ . If $n-k$ is odd or $n-k\in \lbrace 2,n\rbrace$ , we show that all the heights of ISTs are equal to $D(Q_{n,k})+1$ , and thus are optimal. Otherwise, we show that each path from a node to the root in a spanning tree has length at most $D(Q_{n,k})+2$ . In particular, no more than 2.15 percent of nodes have the maximum path length. As a by-product, we improve the upper bound of wide diameter (respectively, fault diameter) of $Q_{n,k}$ from these path lengths.

Independent spanning trees in crossed cubes

Multiple independent spanning trees (ISTs) can be used for data broadcasting in networks, which can provide advantageous performances, such as the enhancement of fault-tolerance, bandwidth, and security. However, there is a conjecture on the existence of ISTs in graphs: If a graph G is n-connected (n⩾1), then there are n ISTs rooted at an arbitrary vertex in G. This conjecture has remained open for n⩾5. The n-dimensional crossed cube CQn is a n-connected graph with various desirable properties, which is an important variant of the n-dimensional hypercube. In this paper, we study the existence and construction of ISTs in crossed cubes. We first give a proof of the existence of n ISTs rooted at an arbitrary vertex in CQn(n⩾1). Then, we propose an O(Nlog2N) constructive algorithm, where N=2n is the number of vertices in CQn.

Independent spanning trees on twisted cubes

Multiple independent spanning trees have applications to fault tolerance and data broadcasting in distributed networks. There are two versions of the n independent spanning trees conjecture. The vertex (edge) conjecture is that any n -connected ( n -edge-connected) graph has n vertex-independent spanning trees (edge-independent spanning trees) rooted at an arbitrary vertex. Note that the vertex conjecture implies the edge conjecture. The vertex and edge conjectures have been confirmed only for n -connected graphs with n ≤ 4 , and they are still open for arbitrary n -connected graph when n ≥ 5 . In this paper, we confirm the vertex conjecture (and hence also the edge conjecture) for the n -dimensional twisted cube T Q n by providing an O ( N log N ) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in T Q n . Moreover, all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n + 1 for any integer n ≥ 2 .

Independent spanning trees on even networks

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. In this paper, we give an algorithm to construct ISTs on even networks, and show that these ISTs are optimal for height and path lengths, and each path in the ISTs has length at most the length of the shortest path+4 in the even network.

Independent spanning trees on folded hyper‐stars

AbstractFault‐tolerant broadcasting and secure message distribution are important issues for numerous applications in networks. It is a common idea to design multiple independent spanning trees (ISTs) as a broadcasting scheme or a distribution protocol for receiving high levels of fault‐tolerance and security. Recently, hyper‐stars were introduced as a competitive model of interconnection network for both hypercubes and star graphs. The class of folded hyper‐stars is a strengthened variation of hyper‐stars obtained by adding additional links to connect complemented nodes. Both hyper‐stars and folded hyper‐stars have been shown to have lower network cost (measured by the product of degree and diameter) than hypercubes, folded hypercubes, and other variants. In this article, we propose an algorithm to construct k + 1 ISTs on a regular folded hyper‐star FHS (2k,k), where the number of ISTs matches the connectivity of FHS(2k,k). In particular, for k &gt; 4, the constructed k ISTs have height 2 k − 2, and the other one has height k + 1. © 2010 Wiley Periodicals, Inc. NETWORKS, 2010

CONSTRUCTING MULTIPLE INDEPENDENT SPANNING TREES ON RECURSIVE CIRCULANT GRAPHS G(2m, 2)

A recursive circulant graph G(N,d) has N = cdm vertices labeled from 0 to N - 1, where d ⩾ 2, m ⩾ 1, and 1 ⩽ c &lt; 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 ± dk ≡ y ( mod N). With the aid of recursive structure, such class of graphs has many attractive features and was considered as a topology of interconnection networks for computing systems. The design of multiple independent spanning trees (ISTs) has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. In the previous work of Yang et al. (2009), we provided a constructing scheme to build k ISTs on G(cdm,d) with d ⩾ 3, where k is the connectivity of G(cdm,d). However, the proposed constructing rules cannot be applied to the case of d = 2. For the integrity of solving the IST problem on recursive circulant graphs, this paper deals with the case of G(2m,2) using a set of different constructing rules. Especially, we show that the heights of ISTs for G(2m,2) are lower than the known optimal construction of hypercubes with the same number of vertices.

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 ) .

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.