Tight Bounds for Parallel Paging and Green Paging

Abstract

In the parallel paging problem, there are p processors that share a cache of size k. The goal is to partition the cache among the processors over time in order to minimize their average completion time. For this long-standing open problem, we give tight upper and lower bounds of Θ(log p) on the competitive ratio with O(1) resource augmentation.A key idea in both our algorithms and lower bounds is to relate the problem of parallel paging to the seemingly unrelated problem of green paging. In green paging, there is an energy-optimized processor that can temporarily turn off one or more of its cache banks (thereby reducing power consumption), so that the cache size varies between a maximum size k and a minimum size k/p. The goal is to minimize the total energy consumed by the computation, which is proportional to the integral of the cache size over time.We show that any efficient solution to green paging can be converted into an efficient solution to parallel paging, and that any lower bound for green paging can be converted into a lower bound for parallel paging, in both cases in a black-box fashion. We then show that, with O(1) resource augmentation, the optimal competitive ratio for deterministic online green paging is Θ(log p), which, in turn, implies the same bounds for deterministic online parallel paging.

Beyond Worst-case Analysis of Multicore Caching Strategies

“…There are very few theoretical guarantees, however, for performance of these algorithms. Furthermore, most existing guarantees on online multicore caching algorithms are nega-tive [31,25], but resource augmentation may be helpful in some cases [1,2].…”

“…We use "multicore caching" because it more accurately reflects the problem studied in this paper. 2 For a cost-minimization problem, an online algorithm has a competitive ratio of c if its cost on any input never exceeds c times the cost of an optimal offline algorithm for the same input (up to an additive constant). 3 LRU is an online caching algorithm that evicts the leastrecently-requested page.…”