Paging and List Update under Bijective Analysis

Abstract

It has long been known that for the paging problem in its standard form, competitive analysis cannot adequately distinguish algorithms based on their performance: there exists a vast class of algorithms which achieve the same competitive ratio, ranging from extremely naive and inefficient strategies (such as Flush-When-Full), to strategies of excellent performance in practice (such as Least-Recently-Used and some of its variants). A similar situation arises in the list update problem: in particular, under the cost formulation studied by Martínez and Roura [TCS 2000] and Munro [ESA 2000] every list update algorithm has, asymptotically, the same competitive ratio. Several refinements of competitive analysis, as well as alternative performance measures have been introduced in the literature, with varying degrees of success in narrowing this disconnect between theoretical analysis and empirical evaluation.In this paper we study these two fundamental online problems under the framework of bijective analysis López-Ortiz, SODA 2007 and LATIN 2008]. This is an intuitive technique which is based on pairwise comparison of the costs incurred by two algorithms on sets of request sequences of the same size. Coupled with a well-established model of locality of reference due to Albers, Favrholdt and Giel [JCSS 2005], we show that LeastRecently-Used and Move-to-Front are the unique optimal algorithms for paging and list update, respectively. Prior to this work, only measures based on average-cost analysis have separated LRU and MTF from all other algorithms. Given that bijective analysis is a fairly stringent measure (and also subsumes average-cost analysis), we prove that in a strong sense LRU and MTF stand out as the best algorithms.

Probabilistic alternatives for competitive analysis

“…Bijective Analysis if there is a bijective mapping φ : S n → S n such that ALG 1 (σ ) ≤ ALG 2 (φ(σ )) for any σ ∈ S n . Very recently, Angelopoulos and Schweitzer [3] applied this idea to paging and showed that LRU is the unique optimal algorithm w.r.t. to Bijective Analysis for a restricted class of sequences exhibiting locality of reference.…”

Stochastic Dominance and the Bijective Ratio of Online Algorithms

“…Such disconnects between the empirical and the theoretical performance evaluation have motivated a substantial line of research on measures alternative to the competitive ratio. Some of the known approaches are: the Max-Max ratio [11]; the diffuse adversary model [32,48,49]; loose competitiveness [47,50]; the random order ratio [30]; the relative worst-order ratio [15,14]; the accommodation function model [17]; and stochastic dominance as well as bijective and average analysis [28,18,19,42,36,34,3,25,4,5]. We refer the reader to the surveys [23,27] for an in-depth discussion of such techniques.…”

“…This latter notion was first introduced in [3] in the context of the paging problem and was shown to be consistent with some natural, "to-be-expected" properties of efficient online algorithms (e.g., the effect of locality of reference as well as lookahead) which competitive analysis fails to yield. For a further discussion of the appealing aspects of bijective analysis, see [3,5]. Definition 1 ( [3]).…”

“…We begin our study of bijective analysis in Section 3, in which we extend the techniques applied in [5] so as to prove the bijective optimality of certain types of greedy algorithms for a much wider class of problems than paging and list update. In particular, we identify some essential conditions under which a certain subclass of greedy-like algorithms (as formally defined in [13]) are optimal.…”

“…In terms of techniques, the transition from exact to approximate bijective analysis necessitates a new approach that combines bijective analysis and amortization arguments. In particular, we note that all previous work that establishes the bijective optimality of a given algorithm [3,4,5,16] is based on inductive arguments which do not immediately carry over to the bijective ratio. For instance [5] crucially exploits the fact that for ρ = 1, if A b ρ·B and B b ρ·C, then A b C. This obviously only holds for ρ = 1, i.e., for optimality.…”