Efficient stack distance computation for priority replacement policies

Abstract

The concept of stack distance, applicable to the important class of inclusion replacement policies for the memory hierarchy, enables to efficiently compute the number of misses incurred on a given address trace, for all cache sizes. The concept was introduced by Mattson, Gecsei, Sluts, and Traiger (Evaluation techniques for storage hierarchies, IBM System Journal, (9)2:78-117, 1970), together with a Linear-Scan algorithm, which takes time O(V) per access, in the worst case, where V is the number of distinct (virtual) items referenced within the trace. While subsequent work has lowered the time bound to O(log V) per access in the special case of the Least Recently Used policy, no improvements have been obtained for the general case. This work introduces a class of inclusion policies called policies with nearly static priorities, which encompasses several of the policies considered in the literature. The Min-Tree algorithm is proposed for these policies. The performance of the Min-Tree algorithm is very sensitive to the replacement policy as well as to the address trace. Under suitable probabilistic assumptions, the expected time per access is O(log2 V). Experimental evidence collected on a mix of benchmarks shows that the Min-Tree algorithm is significantly faster than Linear-Scan, for interesting policies such as OPT (or Belady), Least Frequently Used (LFU), and Most Recently Used (MRU). As a further advantage, Min-Tree can be parallelized to run in time O(log V) using O(V/log V) processors, in the worst case. A more sophisticated Lazy Min-Tree algorithm is also developed which achieves O(√ log V) worst-case time per access. This bound applies, in particular, to the policies OPT, LFU, and Least Recently/Frequently Used (LRFU), for which the best previously known bound was O(V).