Quantifying Competitiveness in Paging with Locality of Reference

Abstract

The classical paging problem is to maintain a two-level memory system so that a sequence of requests to memory pages can be served with a small number of faults. Standard competitive analysis gives overly pessimistic results as it ignores the fact that real-world input sequences exhibit locality of reference. Initiated by a paper of Borodin et al. (J Comput Syst Sci 50:244-258, 1995) there has been considerable research interest in paging with locality of reference. In this paper we study the paging problem using an intuitive and simple locality model that records inter-request distances in the input. A characteristic vector C defines a class of request sequences with certain properties on these distances. The concept was introduced by Panagiotou and Souza (In: Proceedings of 38th annual ACM symposium on theory of computing (STOC), 2006). As a main contribution we develop new and improved bounds on the performance of important paging algorithms. A strength and novelty of the results is that they express algorithm performance in terms of locality parameters. In a first step we develop a new lower bound on the number of page faults incurred by an optimal offline algorithm opt. The bound is tight up to a small additive constant. Technically, the result relies on a new approach of relating the number of page faults to the number of memory hits and amortizing suitably. Based on these expressions for opt's cost, we obtain nearly tight upper and lower bounds on lru's competitiveness, given any characteristic vector C. Furthermore, we compare lru to fifo and fwf. For the first time we show bounds that quantify the difference between lru's performance and that of the other two strategies. The results imply that lru is strictly superior on inputs with a high degree of locality of reference. There exist general input families for which lru achieves constant competitive ratios whereas the guarantees of fifo and fwf tend to k, the size of the fast memory. Finally, we report on an experimental study that demonstrates that our theoretical bounds are very close to the experimentally observed ones. Hence our contributions bring competitive paging again closer to practice.

Relative Worst-order Analysis

“…This gives the desired asymptotic upper bound, showing that WR FWF,LRU = 2k k+1 . There are several other results separating LRU and FWF, e.g., using locality of reference [Becchetti 2004;Albers et al 2005;2015a;Dorrigiv et al 2015;Albers and Frascaria 2018] or competitive analysis parameterized by the attack rate [Moruz and Negoescu 2012]. Moreover, diffuse adversaries separate LRU from both FWF and FIFO [Young 2000] and average analysis combined with locality of reference as modeled in [Albers et al 2005] separates LRU from any other deterministic algorithm [Angelopoulos et al 2019].…”

“…Two of these, diffuse adversaries and average analysis with locality of reference, were mentioned in Section 4.1. Two others use locality of reference, working sets in [Albers et al 2005] and competitive analysis with characteristic vectors in [Albers and Frascaria 2018]. We discuss locality of reference modeled as access graphs in more detail below.…”

The Scheduler is Very Powerful in Competitive Analysis of Distributed List Accessing

“…Theorem 1 There exists a problem, FindValue, in a distributed setting, where there is a randomized algorithm which is 23 16 -competitive against an adaptive, offline adversary, while the best deterministic algorithm is no better than 3 2 -competitive. The first papers applying competitive analysis in a distributed setting [15,17] applied it in a message passing, rather than a shared memory setting.…”

“…Theorem 2 There exists a problem, FindValue, in a distributed setting, where there is a randomized algorithm which is 23 16 -competitive against an adaptive, offline adversary, while the best deterministic algorithm is no better than 3 2 -competitive. This is a counterexample showing that the theorem in [18], making results against adaptive, offline adversaries uninteresting in the sequential online setting, does not necessarily apply to distributed settings.…”

“…Note that multiple requests to the same item in such a sequence are only counted once. Similar definitions of distance have been used for Paging [3], for example.…”