Self-Simulation for the Passive Optical Star

Optical technology offers simple interconnection schemes with straightforward layouts that support complex logical interconnection patterns. The passive optical star (POS), in which communication takes place via optical wavelengths agreed upon between senders and receivers, is often suggested as a platform for implementing the optical network: Logically it offers an all-to-all broadcast capability. We investigate the use of POS optical technology as the communication medium for parallel computing, focusing on the scalability or self-simulation issue, which is a fundamental concern of great importance to the simplicity of algorithm design and program portability. A family of parallel machines is scalable or self-simulating if any member of the family can simulate any larger member with k times as many processors with a slowdown close to k; i.e., scalability states that if a certain efficiency is achievable on a large machine, then it is achievable on any smaller machine as well. A POS is balanced if the number of available wavelengths equals the number of processors. We show that the balanced POS is highly scalable by presenting a universal randomized simulation of a kn-processor balanced POS on an n-processor balanced POS, for arbitrary integers n, k ≥ 1, with an expected slowdown of O(k + log* n). We also describe a deterministic simulation with a slowdown of O(min{k², k + log n}) and show that a slowdown of O(k) is achievable if the communication pattern of each step is known in advance and available for preprocessing. For a certain restricted class of self-simulations we establish matching upper and lower bounds of Θ(k²). We also show that the balanced POS is equivalent to a balanced version of the well-known COLLISION CRCW PRAM, where we now take the balance condition to mean that the number of shared memory cells is the same as the number of processors.