Communication-processor tradeoffs in a limited resources PRAM

We consider a simple restriction of the PRAM model (called PPRAM), where the input is arbitrarily partitioned between a fixed set of p processors and the shared memory is restricted to m cells. This model allows for investigation of the tradeoffs/bottlenecks with respect to the communication bandwidth (modeled by the shared memory size m) and the number of processors p. The model is quite simple and allows the design of optimal algorithms without losing the effect of communication bottlenecks. We have focused on the PPRAM complexity of problems that have Õ(n) sequential solutions (where n is the input size), and where m ≤ p ≤ n. We show essentially tight time bounds (up to logarithmic factors) for several problems in this model such as summing. Boolean threshold, routing, integer sorting, list reversal and k-selection. We get typically two sorts of complexity behaviors for these problems: One type is Õ(n/p+p/m), which means that the time scales with the number of processors and with memory size (in appropriate ranges) but not with both. The other is Õ(n/m), which means that the running time does not scale with p and reflects a communication bottleneck (as long as m < p). We are not aware of any problem whose complexity scales with both p and m (e.g. O(n/√m · p)). This might explain why in actual implementations one often fails to get p-scalability for p close to n.