On Directed Edge-Disjoint Spanning Trees in Product Networks , An Algorithmic Approach

In (Ku et al. 2003), the authors have proposed a construction of edge-disjoint spanning trees EDSTs in undirected product networks. Their construction method focuses more on showing the existence of a maximum number (n1+n2-1) of EDSTs in product network of two graphs, where factor graphs have respectively n1 and n2 EDSTs. In this paper, we propose a new systematic and algorithmic approach to construct (n1+n2) directed routed EDST in the product networks. The direction of an edge is added to support bidirectional links in interconnection networks. Our EDSTs can be used straightforward to develop efficient collective communication algorithms for both models store-and-forward and wormhole.


Introduction
There has been increasing interest over the last two decades in product networks (Day, and Al-Ayyoub 1997;Ku et al. 2003;X and Yang 2007;Imrich et al. 2008;Klavar and Špacapan 2008;Jänicke et al. 2010;Hammack et al. 2011;Chen et al. 2011;Ma et al. 2011;Cheng et al. 2013;Erveš and Žerovnik 2013; Govorčin and Škrekovski 2014).The Cartesian product is a well-known graph operation.When applied to interconnection networks, the Cartesian product operation combines factor networks into a product network.Graph product is an important method to construct bigger graphs, and plays a key role in the design and analysis of networks.A number of spanning trees of a graph are edge-disjoint if no two trees contain the same edge.Edge-Disjoint spanning trees (EDSTs) have many practical applications including enhancing interconnection network fault-tolerance and developing efficient collective communication algorithms in distributed memory parallel computers (Fragopoulo and Akl 1996;Johnsson and Ho 1989;Touzene 2003).In (Ku et al. 2003), the authorshave studied construction of maximum edge-disjoint spanning trees(n 1 +n 2 -1) EDSTs in undirected product network of two graphs, where factor graphs have respectively n1 and n2 EDSTs.The presented construction is more about showing the existence of a maximum number of spanning trees.They did not provide a straight-forward algorithmic way for their construction.In this paper, we propose a new systematic and algorithmic approach to construct (n1+n2) directed rooted edge-disjoint spanning tree in product networks.We assume that the factor graphs are connected graphs and have respectively n1 and n2 EDSTs.Directed rooted edge-disjoint spanning trees have been discussed for different graphs such as the ndimensional hypercube (Johnsson and Ho 1989), k-ary-n-cube (Touzene 2003), star graphs (Fragopoulo and Akl 1996), etc.We assume directed edges: if a and b are two nodes in the graph, the edge (a, b) is different from the edge (b, a).Directed edges support bidirectional links in interconnection networks.The advantage of our method is the direct use of our trees to develop collective communication procedures in product interconnection networks.The remainder of this paper is organized as follows: In Section 2, notations and preliminaries are presented.In Section 3, the construction of edge-disjoint spanning trees in product networks is proposed.In Section 4, we conclude this paper.

Notations and Preliminaries
The Cartesian product G =G 1 ×G 2 of two undirected graphs is the undirected graph G = (V, E), where V and E are given by: V= { <x1, x2> | x1∈V1 and x2∈V2}, and for any u =<x 1 , x 2 > and v = <y 1 , y 2 > in V, (u, v) is an edge in E if, and only if, either In all what follows we consider directed edges in the sense that the edge (u, v) is different from the edge (v, u).

Construction of EDSTs in a Product Network
Consider two graphs having the following properties: the graph G1 contains n1 EDST all rooted at x denoted: X 1 (x), X 2 (x) , … , X n1 (x).Each X i (x) tree is assumed to be formed of an edge (x, x i ), where x i is the i th neighbor of x, and a sub-tree denoted X i (x)/x rooted at x i that spans all the G 1 nodes other than x (Fig. 1.a).The graph G 2 contains n 2 EDST all rooted at y denoted: Y1(y), Y2(y), … , Yn2(y), Each Yj(y) tree is assumed to be formed of an edge (y, y j ), where y j is the j th neighbor of y, and a sub-tree denoted Yj(y)/y rooted at yj that spans all the G2 nodes other than y (figure 1.b).In Fig. 1 (a, b) straight lines correspond to G 1edges and dashed lines correspond to G 2 -edges.In what follows, we fix a specific node <x 0 , y 0 > in G as a desired root for the EDST to be constructed.We denote by <xi, y0>, i = 1,…, n1, the n1 neighbors of <x0, y0>in G reached from <x0, y0> via G1-edges, and by <x0,yj>, j = 1, …, n2, the n2 neighbors of<x0, y0> reached from <x0, y0>via G2-edges.For a given node x in G1 and a given tree Y in G 2 , we denote by <x, Y> the tree in G 1 ×G 2 obtained by fixing the G 1 -component to x and following the edges of tree Y in G 2 .Similarly, <X, y> denotes the tree in G1×G2 obtained by following the edges of a tree X in G1 while the G 2 -component is fixed to node y.

The Special T 1 and T 2 EDSTs for G
We present a construction algorithm for the directed EDSTs in the product graph G denoted T 1 and T 2 .
To illustrate our construction algorithm, we give a complete example of product of two interconnection networks the 3-cube (3 directed rooted EDTS's (Johnsson and Ho 1989)) and a ring with three nodes (a, b and c) (2 directed rooted EDST's).Dark circles represents the root node of the trees and the numbers on the edges

Conclusions
In this paper, we presented a new systematic and algorithmic approach to construct n1+n2 (without using non-tree edges) directed rooted edges-disjoint spanning trees for product networks.The previous work on undirected EDSTs of the product networks (Ku et al. 2003) focuses more on the existence of n 1 +n 2 -1 but did not provide an explicit algorithmic way for their construction.Our n 1 +n 2 EDSTs can be used straight-forward to develop efficient collective communication algorithms for both models store-and-forward and wormhole using bidirectional links.

Figure 1
Figure 1.a.ith EDSTX i (x) rooted at x in G 1 Figure 1.b.ith EDST Y i (y) at y in G 2 and its Xi (x) sub-tree.andits Yj (y)/y sub-tree.

Figure 3 .
Figure 3. Construction of spanning trees T 1 and T 2 .

Figure 4 .
Figure 4. Three EDSTs of the 3-cube and two EDSTs of the ring (3 nodes).
represent the dimension number relative to the 3-cube, see Figs.4 and 5.The trees are directed from the root nodes to leave nodes.