Ring embedding in a family of star graph interconnection networks

Abstract

Graph embedding problem has long been examined for famous interconnection network topologies as a modeling of processor allocation and simulation between parallel architectures in massively parallel processors. As the network topology becomes more complex and larger, the fault tolerance property rises as an important issue. The \emph{star graph interconnection network} has been recognized as an attractive topology that possesses rich performance properties for interconnection network such as diameter, fault-tolerance, simple routing algorithm, and so on. In this thesis, we consider \emph{ring embedding problems} in faulty star graph and its family graphs called the $(n, k)$-star and clustered-star graphs. First, we propose a new ring embedding scheme in a faulty star graph. For this embedding, the \emph{path transition scheme} and the \emph{node borrow technique} are newly proposed in this thesis. The path transition scheme devotes to simplify fault scenarios in the basic component by analyzing path properties. Meanwhile, the node borrow scheme aims at adjusting the path to tolerate the faults in the special positions such as the entry or the exit node in the basic component. Based on these schemes, we show that a ring of length $n! - 2f_v$ can be found in an $n$-star provided that the number of faulty nodes, denoted by $f_v$, is at most $n-3$. Since a star graph is bipartite, our result is optimal in such worst case that all faulty nodes have the same parities, and also superior to the best previous one which makes a ring of length $n! - 4f_v$ under the same fault condition. Moreover, by extending this result into such case that have both node and edge faults, we can construct a ring of length $n! - 2f_v$ in an $n$-star with $f_v$ node faults and $f_e$ edge faults provided that $f_v + f_e \leq n-3$. Second, we examine ring embedding problem in a family of star graphs, called as the $(n, k)$-star and clustered-star graphs. We newly propose a scheme for finding ...