Randomized Throughput-Optimal Oblivious Routing for Torus Networks

In this paper, we study the problem of optimal oblivious routing for 1D and 2D torus networks. 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 enables 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 [19]. When the network radix is odd, W2TURN achieves the minimum average hop count that can be achieved with 2-turn paths while remaining worst-case throughput optimal. When the network radix is even, W2TURN comes very close to achieving the minimum average hop count while remaining worst-case throughput optimal, within just 0.72 percent on a 12 x 12 torus. We also describe another routing algorithm based on weighted random selection of paths with at most two turns called I2TURN and show that I2TURN 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 Weighted Random Direction (WRD). WRD provides a closed-form expression for the optimal distribution of traffic along the minimal and nonminimal directions in a ring topology to achieve minimum average hop count while guaranteeing optimal worst-case throughput. Based on our evaluations, in addition to being worst-case throughput optimal, W2TURN and WRD also perform well in the average case, and outperform the best previously known worst-case throughput optimal routing algorithms with closed-form descriptions in latency and throughput over a wide range of traffic patterns.