A New Worst-Case Throughput Bound for Oblivious Routing in Odd Radix Mesh Network
Torus, mesh, and flattened butterfly networks have all been considered as candidate architectures for on-chip interconnection networks. In this paper, we study the problem of optimal oblivious routing for one of these architecture classes, namely, the torus network. We introduce a new closed-form oblivious routing algorithm called W2TURN that is worst-case throughput optimal for 2D-torus networks. W2TURN is based on a weighted random selection of paths that contain at most two turns. Restricting the maximum number of turns in routing paths to just two results in a simple deadlock-free implementation of W2TURN. In terms of average hop count, W2TURN outperforms the best previously known closed-form worst-case throughput optimal routing algorithm called IVAL . We also provide another routing algorithm based on the weighted random selection of paths with at most two turns called I2TURN and show that it is equivalent to IVAL. However, I2TURN eliminates the need for loop removal at runtime and provides a closed-form analytical expression for evaluating the average hop count. The latter enables us to demonstrate analytically that W2TURN strictly outperforms IVAL (and I2TURN) in average hop count. Finally, we present a new optimal weighted random routing algorithm for rings called WRD (Weighted Random Direction). WRD provides a closed form expression for the the optimal distribution of traffic along the minimal and non-minimal directions in a ring topology to achieve minimum average hop count under maximum worst-case throughput.