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

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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 \(\pi \) defined over all request sequences of a given size such that the cost of A on \(\sigma \) is no more than the cost of B on \(B(\pi (\sigma ))\). 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 et al. (J Comput Syst Sci 70(2):145–175, 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.

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Notes

  1. For an extension of average analysis to problems in which the input contains elements with real-valued weights, see [24]

  2. This is one of the reasons that Albers et al. [3] use the fault rate, instead of overall cost, as a performance measure.

  3. Note that in the context of concave analysis, each continuation must naturally belong in the set \({\mathcal {I}}_h^f\).

  4. We can assume that \(f(2)=2\) since otherwise we are restricted to sequences that contain only one item.

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Acknowledgements

We are grateful to an anonymous referee for the extraordinary effort and care that gave to this paper; the in-depth comments of this referee helped improve substantially the paper’s overall quality, and in particular, the proof of Theorem 2. We also thank Ian Munro for early discussions on alternative measures of performance for online algorithms, as well as Joan Boyar and Kim Larsen for many insightful comments on an earlier version of this paper. This work was the result of an exciting collaboration with our mentor and friend, Alex López-Ortiz, who sadly passed away before this submission was accepted for publication. We would like to dedicate the final paper to his memory.

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Authors and Affiliations

  1. CNRS, Laboratoire d’informatique de Paris 6, LIP6, Sorbonne Université, 75252, Paris, France

    Spyros Angelopoulos

  2. Samsung Research America-Silicon Valley, Mountain View, CA, USA

    Reza Dorrigiv

  3. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Alejandro López-Ortiz

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  1. Spyros Angelopoulos
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  3. Alejandro López-Ortiz
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Correspondence to Spyros Angelopoulos.

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In the conference version of this part of the paper [10], it was claimed that MTF is the unique optimal algorithm, however, we cannot reproduce this proof in this paper. We still conjecture that MTF is uniquely optimal under bijective and concave analysis.

This work is an extended, combined version of two papers: On the separation and equivalence of paging strategies, in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 229–237, and List Update with Locality of Reference, in Proceedings of the Eighth Latin American Theoretical Informatics Symposium, 2008, pp. 399–410.

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Angelopoulos, S., Dorrigiv, R. & López-Ortiz, A. On the Separation and Equivalence of Paging Strategies and Other Online Algorithms. Algorithmica 81, 1152–1179 (2019). https://doi.org/10.1007/s00453-018-0461-2

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