On the strong distance problems of pyramid networks

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Abstract

Suppose G=(V,E) is a graph and D=(V,F) is a strong digraph of G. Let u and v be two vertices of D. The strong distance sd(u,v) is the minimum size of the strong subdigraph of D containing u and v, and the strong eccentricity se(u) is the maximum strong distance sd(u,v) for all vertex v in D. The strong radius and the strong diameter of D are defined as the minimum and maximum strong eccentricity se(u) for all u in D, respectively. In this paper, we present a lower bound of strong diameter (radius) for any strong digraph. Further, we propose a better upper bound of the strong diameter for any Hamiltonian strong digraph. Moreover, we study the strong distance problems on pyramid networks, PM[n]. We give a lower bound to SDIAM(PM[n]) and SRAD(PM[n]). Finally, we conclude the exact value of sdiam(PM[n]), as well as an upper and a lower bound of srad(PM[n]).

Introduction

In the design of networks, one of the most important topics is their reliability. In practice, we are often interested in the intercommunication between a pair of nodes. When the edges between any two connected nodes are one-way, this problem becomes difficult. In order to achieve better performance of communications, it is necessary to consider how to optimize the routing of two directed paths between a pair of nodes. This reflects one important aspect of the strong distance problems of the interconnection networks which has not been studied for most network topologies.

In this paper, we are interested in the strong distance problems on an important network, the pyramid network. The pyramid network is one of the important structures in parallel and network computing [1], [2], and image processing [3]. A lot of parallel algorithms are efficiently implemented on a pyramid network, and each processor, or workshop, acting as a node in the pyramid network in parallel and network computing. In image processing, the pyramid networks are used as both hardware architectures and software structures.

We begin by introducing some definitions of the digraph (directed graph) D=(V,F), and they can also be referred to [4], [5], [6], [7], [8], [9], [10]. Recall that for an undirected graph G and x,yV(G), distance of x and y, denoted by d(x, y), is defined as min{∣E(P)∣ : P is a (x, y)-path}; the eccentricity of x, denoted by e(x), is defined as max{d(x, y) : y  V(G)}; the diameter of G, denoted by diam(G), is defined as max{e(x) : x  V(G)} and the radius of G, denoted by rad(G), is defined as min{e(x) : x  V(G)}. For a digraph D, the size of D is the number of edges of D, which is denoted by ∣D∣. And the order of D is the number of vertices of D, which is denoted by ∣V(D)∣. A digraph D′ is a subdigraph of D if V(D)V(D),F(D)F(D). For two digraphs D1 and D2, D1D2(D1D2) is a digraph whose vertex set is the intersection(union) of the vertex set of D1 and D2, and edge set is the intersection(union) of the edge set of D1 and D2. A graph G is an underlying graph of D if we can obtain G by replacing every directed edge of D with an undirected edge. Conversely, with any graph G, we can obtain a digraph from G by specifying each edge of G an order of its ends. Such a specification is called an orientation of G. An oriented graph is an orientation of a simple graph. A digraph D is called strong digraph if for every pair u, v of distinct vertices of D, there exist both a directed (u, v)-path and a directed (v, u)-path. A digraph H is a strong subdigraph of D if H is a subdigraph of D and H is a strong digraph. A digraph D is called a strong orientation of G if D is an oriented graph of G and D is a strong digraph. A strong orientated graph is a strong orientation of a simple graph. The distance problems of digraphs has been studied widely as [6], [8], [11], [12]. In [6], Chartrand et al. defined strong distance sd(u, v) (written sdD(u,v) when reference to D is needed) between two vertices u and v in a strong digraph is defined as the minimum size of a strong subdigraph of D containing u and v. Besides, Chartrand et al. defined the strong eccentricity se(u) of a vertex u in a strong oriented graph D as max{sd(u,v)|vV(D)}. The strong diameter sdiam(D) of D is defined as max{se(u)|uV(D)} and the strong radius srad(D) of D is defined as min{se(u)|uV(D)}.

In [8], Lai et al. defined lower (upper, resp.) orientable strong radius, srad(G) (SRAD(G), resp.), of G as min (max, resp.) {srad(D)∣D is a strong orientation of G}. And lower (upper, resp.) orientable strong diameter, sdiam(G) (SDIAM(G), resp.), of G as min (max, resp.) {sdiam(D)∣D is a strong orientation of G}.

A pyramid network is a hierarchy structure based on meshes. It is a desirable network topology used as both software data-structure and hardware architecture and be studied widely in recent years [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. At first, we introduce Cartesian product G1×G2 of two graph G1 and G2. The vertex set of V(G1×G2) is {(u,v)|uV(G1),vV(G2)} and the edge set of E(G1×G2) is {((u,v),(u,v))|u=u and vvE(G1), or v=v and uuE(G2)}. The mesh graph, denote by Mm,n, is Pm×Pn.

The n-dimensional pyramid network (also called a pyramid), denoted by PM[n], is defined as the vertex set of V(PM[n]) is {(k;x,y)|0kn,1x,y2n}. A node, (k;x,y)V(PM[n]), is said to be a node at level k. All the nodes in level k form a mesh M2k,2k. And for 0k<n, the node (k;x,y)V(PM[n]) is also connected to (k+1;2x-1,2y-1),(k+1;2x-1,2y),(k+1;2x,2y-1) and (k+1;2x,2y). The node B=(k-1;x,y) is said to be the parent of A=(k;x,y), denoted by P(A) = B, if (k-1;x,y)(k;x,y)E(PM[n]). Conversely, A is a child of B. Let Pi(A) (Ci(A)) denote the ith ancestor (descendant) of a node A, which is defined as follows:

  • 1.

    i = 0, P0(A)(C0(A))=A.

  • 2.

    i = 1, P1(A)=P(A)(C1(A)=C(A)) is simply the parent (a child) of A.

  • 3.

    i  2, Pi(A)=P(Pi-1(A)) (Ci(A)=C(Ci-1(A))) is the parent (a child) of Pi-1(A) (Ci-1(A)).

The node (0; 1, 1) is called the apex of PM[n]. It is easy to see that the radius and diameter of PM[n] is n and 2n, respectively [9]. Fig. 1 illustrates the PM[2].

In this paper, we use the graph and network interchangeably. Also, we use the vertex and node interchangeably. Chartrand et al. discuss some properties of a strong oriented graph in [6], it shows that the underlying graph of a strong oriented graph is necessarily 2-edge connected. They also discuss the strong radius (diameter) of strong tournaments in [5]. In [8], [24], they discuss the values of srad(G), sdiam(G), SRAD(G) and SDIAM(G) when G is equal to complete a bipartite graph or hypercube. In [25], Juan and Huang discuss bound on srad(G), sdiam(G), SRAD(G) and SDIAM(G), and an upper bound of srad(G1×G2) and sdiam(G1×G2) where G1×G2 means the Cartesian product of two graphs G1 and G2. In [26], Dankelmann et al. discuss the upper bound on strong radius and strong diameter. In [27], the strong distance problems of toroidal and semitoroidal mesh graphs were discussed.

This paper is organized as follows. In Section 2, we study the lower bounds of srad(G) and sdiam(G) for general graph G. Then we discuss the exact values of sdiam(PM[n]) and srad(PM[n]). In Section 3, we present an upper bound of the strong diameter of some special strong digraphs. And give an lower bound for SDIAM(PM[n]) and SRAD(PM[n]). The conclusion is given in Section 4.

Section snippets

sdiam(PM[n]) and srad(PM[n])

We define some notations at first. A edge-container EC(x,y) between two distinct nodes x and y in G is a set of edge-disjoint paths between x and y. The width of EC(x,y), written as w(EC(x, y)), is its cardinality. The size of EC(x,y), written as s(EC(x, y)), is the total number of edges in every path of EC(x,y). The w-wide total-distance between two nodes x and y, denoted by tdw(x,y), is min{s(EC(x,y))|EC(x,y)is a edge-container with widthw}. The w-wide total-diameter of G, written as tdw(G), is

SDIAM(PM[n]) and SRAD(PM[n])

In this section, we present an upper bound of the strong diameter for any Hamiltonian strong directed graphs, and a lower bound of SDIAM(PM[n]) and SRAD(PM[n]). Given a digraph D, the directed girth g(D) is defined as length of a shortest directed cycle in D. In [26], Dankelmann et al. discuss the relationship between the strong diameter of a strong digraph D and the directed girth of D. They have the following results:

Theorem 3.1

see [26]

If D is a strong digraph of order n and directed girth g  2, then sdiam(D)(n

Conclusion

In this paper, we study some properties of strong distance problems. We defined trw(G) and tdw(G), and use that to give a lower bound for srad(D) and sdiam(D) for any strong orientation of G. After that, we discussed the strong distance problems on pyramid networks. We gave the exact value for sdiam(PM[n]), and gave an upper and a lower bound for srad(PM[n]). The gap between them is only 1. Indeed, it can be shown that the upper bound is equal to srad(PM[n]) when 1n3, which leads us to the

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