Double-loop networks are widely used in computer networks. In this paper, we present an optimal message routing algorithm and an optimal fault-tolerant message routing algorithm for weighted bidirectional double-loop networks. The algorithms presented are novel, and they do not use routing tables. After a precalculation of O(log N) steps to determine network parameters, the algorithms can route messages using constant time at each node along the route. The algorithm presented can route messages in the presence of up to three faulty nodes or links. The fault-tolerant routing algorithm guarantees an optimal route in the presence of one node failure.

A multiple bus network (MBN) connects a set of processors via set of buses. Two important parameters of an MBN are its loading (largest number of connections on a bus) and its degree (largest number of connections to a processor). These parameters determine the cost, speed, and implementability of the MBN. The smallest degree that any useful MBN can have is 2. In this paper, we study the relationship between running time, degree, and loading of degree-2 MBNs running a fundamental class of algorithms called binary tree algorithms. (A binary tree algorithm reduces 2/sup n/ inputs at the leaves of a balanced-binary tree to a single result at the root of the tree.) Specifically, we establish a nontrivial /spl Omega/(n/logn) loading lower bound for any degree-2 MBN running a 2/sup n/ input binary tree algorithm optimally in n steps. We show that this bound does not hold if the restriction on the degree or the running time is relaxed. That is, optimal-time, degree-3, constant loading MBNs and suboptimal-time, degree-2, constant loading MBNs exist for binary tree algorithms. We also derive a lower bound on the additional time (beyond the optimal) needed to run binary tree algorithms on a degree-2, loading-L MBN, for any L>3.

We consider a class of algorithms called fan-in algorithms, with numerous applications in problems involving semigroup operations. We present a multiple bus network (MBN) that runs any fan-in algorithm in optimal number of steps. The degree and loading of this MBN are each 3. We prove that the product of the degree and loading of any MBN that runs a fan-in algorithm in optimal time is at least 9. This establishes the proposed MBN to be optimal.