Self-Simulation for the Passive Optical Star

Optical technology o ers simple interconnection schemes with straightforward layouts that support complex logical interconnection patterns. The Passive Optical Star (pos) is often suggested as a platform for implementing the optical network: Logically it o ers an all-to-all broadcast capability. We investigate the use of pos optical technology as the communication medium for parallel computing. In particular, a feature of parallel models which is extremely important for the simplicity of algorithm design and program portability is the scalability property or self-simulation capability. It states that when a computation achieves a certain speedup on a large machine with many processors, then it achieves a similar speedup on any smaller machine (relative to the number of processors). We show that the pos is indeed scalable, namely, we present a randomized algorithm for an n-processor pos that does not assume global knowledge and that simulates a kn-processor pos with a slowdown of O(k + log n). We also analyze direct algorithms, namely, algorithms that send messages directly from the original sender to the nal destination. We prove that k2 is the exact complexity if we allow the use of the original set of frequencies, and k3 is an upper bound if the set of frequencies is restricted too. On the way to our self-simulation result we show that the pos computational model is equivalent to the well known crcw collision pram with shared memory of linear size, hence the same results hold for the latter model as well. 1