If uzi*+!uzi* = i&*+1V2?*, it requires two steps in P. Otherwise, by the definition of iâ it requires a single edge in P. If uzj+luzj - v2j+1v2jl after identifying.
A New Shortest Path Routing Algorithm and Ernbedding Cycles of Crossed Cube Chien-Ping Changl, Ting-Yi Sung2 and Lih-Hsing Hsul Department of Computer and Information Science National Chiao Tung University Hsinchu, Taiwan 30050, R.O.C.' Institute of Information Science Academia Sinica Taipei, Taiwan 11529, R.0.C.2 Abstract
In order to evaluate the performance of a network topology, we can consider the following measures: shortest path routing complexity, vertex connectivity, diameter and embedding of cycles. Efe presented a shortest path routing algorithm of crossed cubes in [l],which generated one shortest path for any pair of vertices in O ( n 2 )time. In this paper, we define a new distance measure which enables us to find more shortest paths for any pair of vertices in O ( n ) time. The vertex connectivity (simply abbreviated as connectivity) of a network G = ( V , E ) ,denoted by K ( G )or K , is the minimum number of vertices whose removal leaves the remaining graph disconnected or trivial. It follows from Menger's theorem that there always exist K internally vertex-disjoint (abbreviated as disjoznt) paths between any two vertices. Disjoint paths between a pair of vertices contribute t o multipath communication between these two vertices and provide alternative routes in the case of vertex or link failures. Thus large conenctivity is preferred. It has been shown in [1, 41 that K ( C Q ~=) K ( & , ) = n. The problem of simulating one network by another can be modeled as a graph emdedding problem. Embeddings of complete binary trees and cycles into crossed cubes were pxesented in [3] and [6], respectively. In [6], the authors constructed one type of cycles for an arbitrary length whereas we construct various types of cycles in this paper. The rest of this paper is organized as follows. Section 2 summarizes some known results on crossed cubes and introduces aotation used in this paper. In section 3 we give a nlzw shortest path routing algorithm which runs in O ( n ) time. Embedding of cycles into crossed cubes by constructing various types of cycles is presented in section 4. Finally, we make concluding remarks in section 5.
A n n-dzmensaonal crossed cube, CQn, zs a varzataon of hypercubes. In thzs paper, we gave a new shortest path routzng algorzthm based on a new dastance measure defined herean. In comparzson wath Efe's algorzthm whzch generates one shortest path an O(n2) tame, our algorathm can generate more shortest paths zn O(n) tame. Furthermore, we show that CQn zs a pancyclzc network and we construct varaous types of cycles of an arbztrary length at least four
1
Introduction
Network topology is a crucial factor for interconnection networks since it determines the performance of a network. Many interconnection network topologies have been proposed in literature for connecting hundreds or thousands of processing elements. Network topology is always represented by a graph where vertices represent processors and edges represent links between processors. Among these topologies, the binary n-cube (or binary hypercube), denoted by Q n , IS one of popular topologies. In a binary n-cube with N = 2n vertices and links, we say that the link from vertex x = x n - l x n - - 2 . . ' 2 1 x 0 t o vertex y = yn-1yn-2 .ylyo spans dimension a if and only if zt # yi and xJ = y j for all j # i. We call this link the i-th link a t vertex x or y. However, a binary n-cube does not make the best use of its hardware in the following sense: given N = 2" vertices and links, it is possible to fashion networks with lower diameters than that of Q n . One such topology is the crossed cube, which was first proposed by Efe El]. An n-dimensional crossed cube, denoted by CQn, has the same number of vertices and links as Qn and is derived from changing the connection of some hypercube links. I t has a diameter of [ 9 1 an , improvement of approximately a factor of 2 , in a trade-off of reducing high degree of symmetry in Q n . Other properties of crossed cubes are also studied in literature [1, 2 , 3, 4 , 61. '
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1087-4089197 $10.00 0 1997 IEEE
2
Preliminaries; and Notation
Let G be a graph. We use V ( G ) and E ( G ) to denote the vertex set and the edge set of G,
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respectively. Let x and y be two vertices. We use d c ; ( z , y ) t o denote the shortest distance in G between x and y. To define crossed cubes, we first introduce the notation of "pair related " . Let R = {(00,00),(10,10),(01,11),(11,01)}. Two binary strings x = xlxo and y = y l y o are pair related if and only if ( x ,y) E R.
Similarly, we can contract those vertices in CQak+l with the same prefix of length three into a vertex and obtain a graph with eight vertices. Again, this eightvertex graph is isomorphic t o CQ3 as illustrated in Figure 2(b). We can also obtain the following observations for any two vertices u , v in CQn with n odd and n 2 3 [l]:
Definition 1 A n n-dimensional crossed cube CQn is recursively constructed as follows: CQ1 is a complete graph with two vertices labeled b y 0 and 1, respectively. CQn consists of two identical ( n - 1)dimenszonal crossed cubes C Q i - , and CQh-l. The vertex U = 0un-2...uo E V(CQ;-,) and the vertex v = 1wn1~...vo E V(CQA-,) is an edge in CQn if and only zf
( b l ) If p3(u) E p ~ ( v )then , U and v are in a subgraph isomorphic t o CQn-3.
(i)
un-2
= vn-2 if n is even, and
(ii) f o r 0 5 i
