Optimal on-line computation of stack distances for MIN and OPT

The replacement policies known as MIN and OPT are optimal for a two-level memory hierarchy. The computation of the cache content for these policies requires the off-line knowledge of the entire address trace. However, the stack distance of a given access, that is, the smallest capacity of a cache for which that access results in a hit, is independent of future accesses and can be computed on-line. Off-line and on-line algorithms to compute the stack distance in time O(V) per access have been known for several decades, where V denotes the number of distinct addresses within the trace. The off-line time bound was recently improved to O(√V log V). This paper introduces the Critical Stack Algorithm for the online computation of the stack distance of MIN and OPT, in time O(log V) per access. The result exploits a novel analysis of properties of OPT and data structures based on balanced binary trees. A corresponding Ω(log V) lower bound is derived by a reduction from element distinctness; this bound holds in a variety of models of computation and applies even to the off-line simulation of just one cache capacity.