A Characterization of Temporal Locality and Its Portability across Memory Hierarchies

This paper formulates and investigates the question of whether a given algorithm can be coded in a way efficiently portable across machines with different hierarchical memory systems, modeled as a(x)-HRAMs (Hierarchical RAMs), where the time to access a location x is a(x). The width decomposition framework is proposed to provide a machine-independent characterization of temporal locality of a computation by a suitable set of space reuse parameters. Using this framework, it is shown that, when the schedule, i.e. the order by which operations are executed, is fixed, efficient portability is achievable. We propose (a) the decomposition-tree memory manager, which achieves time within a logarithmic factor of optimal on all HRAMs, and (b) the reoccurrence-width memory manager, which achieves time within a constant factor of optimal for the important class of uniform HRAMs. We also show that, when the schedule is considered as a degree of freedom of the implementation, there are computations whose optimal schedule does vary with the access function. In particular, we exhibit some computations for which any schedule is bound to be a polynomial factor slower than optimal on at least one of two sufficiently different machines. On the positive side, we show that relatively few schedules are sufficient to provide a near optimal solution on a wide class of HRAMs.