Fast, efficient mutual and self simulations for shared memory and reconfigurable mesh

This paper studies relations between the parallel random access machine (PRAM) model, and the reconfigurable mesh (RMESH) model, by providing mutual simulations between the models. We present an algorithm simulating one step of an (n lg lg n)-processor CRCW PRAM on an n × n RMESH with delay O(lg lg n) with high probability. We use our PRAM simulation to obtain the first efficient self-simulation algorithm of an RMESH with general switches: An algorithm running on an n × n RMESH is simulated on a p × p RMESH with delay O((n/p)² + lg n lg lg p) with high probability, which is optimal for all p ≤ n/√lg n lg lg n. Finally, we consider the simulation of RMESH on the PRAM. We show that a 2 × n RMESH can be optimally simulated on a CRCW PRAM in Θ(α(n)) time, where α(·) is the slow-growing inverse Ackermann function. In contrast, a PRAM with polynomial number of processors cannot simulate the 3 × n RMESH in less than Ω(lg n/lg lg n) expected time.