Competitive Analysis of Move-to-Front-or-Middle (MFM) Online List Update Algorithm

TL;DR

This paper analyzes the Move-to-Front-or-Middle (MFM) online list update algorithm using competitive analysis, revealing its competitiveness bounds and addressing open questions about offline algorithms' role in online algorithm analysis.

Contribution

It provides the first bounds on MFM's competitive ratio, showing it is 2-competitive with dynamic offline algorithms but not with static offline algorithms.

Findings

MFM is 2-competitive with dynamic offline algorithms.

MFM is not 2-competitive with static offline algorithms.

Theoretical bounds open new research directions in online list update algorithms.

Abstract

The design and analysis of efficient algorithms with the knowledge of current and past inputs is a non-trivial and challenging research area in computer science. In many practical applications the future inputs are not available to the algorithm at any instance of time. So the algorithm has to make decisions based on a sequence of inputs that are in order and on the fly. Such algorithms are known as online algorithms. For measuring the performance of online algorithms, a standard measure, known as competitive analysis, has been extensively used in the literature. List update problem is a well studied research problem in the area of online algorithms since last few decades. One of the widely used deterministic online list update algorithm is the Move-To-Front (MTF) algorithm, which has been shown to be 2-competitive with best performance in practical real life inputs. In this paper we analyse the Move-to-Front-or-Middle (MFM) algorithm using competitive analysis by addressing one of an open question raised by Albers that whether dynamic offline algorithm can be used in finding the competitiveness of an online algorithm? Move-To-Front-or-Middle (MFM) was experimentally studied and observed to be performing better than MTF algorithm by using the Calgary Corpus and Canterbury Corpus data set. However, it is interesting and challenging to find the lower bound and upper bound on the competitive ratio of MFM algorithm. We make a first attempt to find the competitiveness of MFM algorithm. Our new results show that MFM is not 2-competitive with respect to static optimum offline algorithm, whereas it is 2-competitive with respect to dynamic optimum offline algorithm. Our new theoretical results may open up a new direction of research in the online list update problem by characterising the structure of competitive and non competitive deterministic online algorithms.