Construction of optimal independent spanning trees on folded hypercubes☆
Introduction
A tree T in a graph G is called its spanning tree if T contains all vertices of G. A rooted tree is a tree with its one vertex r chosen as root. A set of k(⩾2) spanning trees rooted at the same vertex on a graph is said to be edge-disjoint if any two trees of the set have no common (directed) edges. The problem of constructing edge-disjoint spanning trees has been considered for arrangement graphs [23], folded hypercubes [13], locally twisted cubes [15] and twisted cubes [33], etc.
Independent spanning trees on a graph have stronger properties than those of edge-disjoint spanning trees. Let T be a spanning tree of a graph G rooted at a vertex r. For a given vertex x ∈ V(T)⧹{r}, let P(T, x) denote the unique path from r to x in T. A set of k(⩾2) spanning trees T1, T2, … , Tk on a graph G rooted at a vertex r is said to be independent if for any vertex x ∈ V(G)⧹{r} the k paths P(T1, x), P(T2, x), … , P(Tk, x) are internally disjoint (any two paths P(Ti, x) and P(Tj, x) have no common vertex except two end-vertices x and r for 1 ⩽ i < j ⩽ k).
The design of independent spanning trees (ISTs) on graphs has important applications in the reliable broadcasting as well as distributing secure messages [1], [2], [17], [27]. If a spanning tree rooted at the source node in a network is viewed as a broadcast channel for data communication, then fault-tolerant broadcasting can be achieved by sending k copies of a message along k ISTs on the network provided that there are at most k − 1 faulty nodes (other than the root) and/or faulty edges in the network. If a message at the source node is separated into k different parts, by sending the k parts along k ISTs on the network, then the message is secure in the message distributing.
In this regard, the problem of constructing ISTs on graphs has received much attention. However, the problem is very tough for arbitrary graphs. It was conjectured that for any k-connected graph there exist k ISTs rooted at its any vertex [41]. The conjecture has been proved true for k-connected graphs with k ⩽ 4 [8], [9], [17], [41] and some classes of graphs or digraphs (in particular, interconnection networks) such as planar graphs [16], product graphs [26], chordal rings [18], [37], star networks [27], de Bruijn and Kautz digraphs [12], torus networks [29], recursive circulant graphs [38], [39], Cartesian product of complete graphs [6], folded hyper-stars [34], hypercubes [28], [40], folded hypercubes [35], [36], locally twisted cubes [24], twisted cubes [30], crossed cubes [7], parity cubes [31], even networks [19], and odd graphs [20].
The n-dimensional hypercube Qn is one of the most popular and efficient interconnection networks due to its many excellent properties. There is a large amount of literature on the properties of hypercubes and their applications. As an important variant of Qn, the n-dimensional folded hypercube FQn (n ⩾ 2), proposed by El-Amawy and Latifi [10], is the graph obtained from Qn by adding an edge between any pair of vertices with complementary addresses. It is known that FQn is vertex-transitive and edge-transitive and has many properties superior to Qn. As examples, its connectivity is n + 1; its diameter is ⌈n/2⌉, about half the diameter of Qn; and there exists a cycle of each length l with n ⩽ l ⩽ 2n if n(⩾4) is even. In this regard, there have been increasing studies on folded hypercubes recently [4], [5], [11], [14], [21], [22], [25], [32], [42].
In [13] Ho proposed an algorithm for constructing n + 1 edge-disjoint spanning trees on the folded hypercube FQn. Recently, Yang et al. [36] showed that the n + 1 edge-disjoint spanning trees on FQn constructed by Ho are indeed independent and the height of each spanning tree is n. In this paper, we propose an algorithm for constructing n + 1 optimal ISTs on the folded hypercubes FQn in the sense that there is a shortest path between the only child of the root and any other vertex in each spanning tree, so the height of each spanning tree is ⌈n/2⌉ + 1. Therefore, both the average path length and the height of each spanning tree are optimal. Moreover, the algorithm runs in time O((n + 1)N) and can be parallelized to run by using N = 2n processors on FQn in time O(n).
The rest of this paper is organized as follows: Section 2 gives some preliminaries. In Section 3 an algorithm for constructing n + 1 optimal ISTs on FQn is proposed. In Section 4 we prove the correctness of the algorithm. Conclusion remarks are given in Section 5.
Section snippets
Preliminaries
In this section some preliminaries are given. We follow [3] for graph-theoretical terminology and notation. Throughout this paper, a graph G = (V, E) means a simple graph, where V = V(G) is the vertex-set and E = E(G) is the edge-set of the graph G. The number of the edges contained in a path P is called the length of the path P. The length of a shortest path connecting two vertices x and y in a graph G is called the distance between x and y in G and denoted by dG(x, y). Let D(G) = max{dG(x, y): x, y ∈ V(G
Algorithm for constructing n + 1 optimal ISTs on FQn
In this section, we propose an algorithm for constructing n + 1 optimal independent spanning trees rooted at the vertex O in the folded n-cube FQn: T0, T1, … , Tn−1 and T∗. To construct these spanning trees, we are to determine the parent of each vertex x other than the root O in each spanning tree. Our algorithm only depends on the binary string of the vertex x, so it can be easily implemented in parallel or distributed systems. Algorithm 1Input: any vertex x = δn−1, δn−2, … , δ0 ∈ V(FQn)⧹{O} Output: f0(x), f1(x),
Proof of correctness of Algorithm
In this section, we are to show that T0, T1, … , Tn−1 and T∗ constructed by Algorithm are n + 1 optimal ISTs on the folded n-cube FQn. We will show the following three theorems. Theorem 1 For each ω ∈ I ∪ {∗}, Tω constructed by Algorithm is an optimal spanning tree in the folded n-cube FQn. Proof 2 Given any vertex x = δn−1δn−2… δ0 ∈ V(FQn)⧹{O}. Assume ω = i ∈ I. By Algorithm, fi(εi) = O. Let x ≠ εi. By Proposition 1, d(fi(x), εi) = d(x, εi) − 1. Let T′ be a subgraph in the folded n-cube FQn, whose vertex set is V(T′) = V(FQn)⧹{O} and whose
Conclusion remarks
Yang et al. [36] proposed an algorithm for constructing n + 1 ISTs on the folded n-cube FQn. While designing their algorithm, they did not consider the relation between the Hamming distance h(x, O) and the distance d(x, O) of a pair of vertices x and O in FQn (see Lemma 1). Thus, the height of each spanning tree constructed in [36] is not optimal. In this paper, we propose an algorithm for constructing n + 1 optimal ISTs on FQn in the sense that there is a shortest path between the only child of the
Acknowledgement
The author thanks the anonymous referees for their review comments that help to improve the original manuscript.
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The work was supported by NSF of Fujian Province in China (Nos. 2010J01354 and 2011J01025).