Communication - Processor Tradeo s in Limited Resources

We consider a simple restriction of the PRAM model (called PPRAM), where the input is arbitrarily partitioned between a xed set of p processors and the shared memory is restricted to m cells. This model allows for investigating the tradeoos/ 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 loosing the eeect of communication bottlenecks. We have focused on the PPRAM complexity of problems that have O(n) sequential solutions (where n is the input size), and where m p n. We show tight time bounds 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: Either ~ O(n=p+p=m) which means that the time scales with the number of processors and with memory size (in appropriate range) but not with both. The other is a ~ O(n=m) which does not scales with p and reeects 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 p mp)). This might explain why in actual implementations one often fails to get p-scalability for p close to n.