Fault-Tolerant Routing in DeBruijn Comrnunication Networks

A class of communication networks which is suitable for "multiple processor systems" was studied by Pradhan and Reddy. The underlying graph (to be called Shift and Replace graph or SRG) is based on DeBruijn digraphs and is a function of two parameters r and m. Pradhan and Reddy have shown that the node-connectivity of SRG is at least r. The same authors give a routing algorithm which generally requires 2m hops if the number of node failures is ≤(r -1). In this paper we show that the node-connectivity of SRG is (2r - 2). This would immediately imply that the system can tolerate up to (2r - 3) node failures. We then present routing methods for situations with a certain number of node failures. When this number is ≤(r - 2) our routing algorithm requires at most m + 3 + logr m hops if 3 + logr m ≤m. When the number of node failures is ≤(2r - 3) our routing algorithm requires at most m + 5 + logr m hops if 4 + logr m ≤ m. In all the other situations our routing algorithm requires no more than 2m hops. The routing algorithms are shown to be computationally efficient.