Fault Tolerance on Star Graphs

A multiprocessor system consists of a set of processing units and each of them has its own local memory. The processing units in a multiprocessor system are linked in some topology. What we are interested in is a topology proposed by Akers et al. [2,3], called star graph. An n-dimensional star graph can be represented by S,, = (V,,, E,,), where V,, consists of n! nodes in which each node is identified by one of the permutations of { 1 , 2 , . , n ) . For any a,b E V,,, (a ,b) E E,, if and only if there exists some i , 2 5 i L n, such that a and b are identified as (p1 p2 ... pi-1 pi pi+l .. . p,,) and (pi p2 * * pi-1 p i pj+l p n ) respectively. We write it as gi(a) = b. The notation SL represents an m-substar in S,, in which the (m + 1)th position symbol is 6. We illustrate S4 in Figure 1. The star graph is comparable to the hypercube in many aspects. For example, both of them are edge symmetric, node symmetric, strongly hierarchical, bipartite and optimally fault tolerant. But star graphs offer a better degree and diameter than hypercubes. So the star graph structure has been considered as an attractive alternative to the hypercube structure.