On the Separation and Equivalence of Paging Strategies and Other Online Algorithms

Abstract

We introduce a new technique for the analysis of online algorithms, namely bijective analysis, that is based on pair-wise comparison of the costs incurred by the algorithms. Under this framework, an algorithm A is no worse than an algorithm B if there is a bijection π defined over all request sequences of a given size such that the cost of A on σ is no more than the cost of B on B(π(σ)). We also study a relaxation of bijective analysis, termed average analysis, in which we compare two algorithms based on their corresponding average costs over request sequences of a given size. We apply these new techniques in the context of two fundamental online problems, namely paging and list update. For paging, we show that any two lazy online algorithms are equivalent under bijective analysis. This result demonstrates that, without further assumptions on characteristics of request sequences, it is unlikely, or even undesirable, to separate online paging algorithms based on their performance. However, once we restrict the set of request sequences to those exhibiting locality of reference, and in particular using a model of locality due to Albers, Favrholdt, and Giel [JCSS 2005], we demonstrate that Least-Recently-Used (LRU) is the unique optimal strategy according to average analysis. This is, to our knowledge, the first deterministic model to provide full theoretical backing to the empirical observation that LRU is preferable in practice. Concerning list update, we obtain similar conclusions, in terms of the bijective comparison of any two online algorithms, and in terms of the superiority (albeit not necessarily unique) of the Move-To-Front (MTF) heuristic in the presence of locality of reference.

List Update with Locality of Reference

“…This result parallels the equivalence of all lazy paging algorithms under Bijective Analysis as shown in [6]. Proof.…”

“…We choose to employ these measures since they reflect certain desirable characteristics for comparing online algorithms: they allow for direct comparison of two algorithms without appealing to the concept of the "optimal" cost (see [6] for a more detailed discussion), and they do not evaluate the performance of the algorithm on a single "worst-case" request, but instead use the cost that the algorithm incurs on each and all request sequences. These two measures have already been successfully applied in the context of the paging problem [6].…”

“…Definition 2. [6] We say that an on-line algorithm A is no worse than an online algorithm B according to Average Analysis if there exists an integer n 0 ≥ 1 so that for each n ≥ n 0 , I∈In A(I) ≤ I∈In B(I). We denote this by A a B.…”

“…For example, for some problems OPT is too powerful, causing all on-line algorithms to seem equally bad. Certain alternative measures allow direct comparison of on-line algorithms, for example the Max-Max Ratio [9], Relative Worst Order Ratio [11], Bijective Analysis and Average Analysis [6]. These measures have been applied mostly to the paging problem as well as some other on-line problems.…”

Quantifying Competitiveness in Paging with Locality of Reference

“…These include (a) the max/max ratio [6], (b) bijective and average analysis [2,3], (c) the relative worst-order ratio [7,8], (e) relative interval analysis [9,14] and (f) parametrized analysis [11]. In a bijective analysis two algorithms alg 1 and alg 2 are compared on permutations of the same requests.…”