Online algorithms

Abstract

This chapter reviews fundamental concepts and results in the area of online algorithms. We first address classical online problems and then study various applications of current interest. Online algorithms represent a theoretical framework for studying problems in interactive computing. They model, in particular, that the input in an interactive system does not arrive as a batch but as a sequence of input portions and that the system must react in response to each incoming portion. Moreover, they take into account that at any point in time future inputis unknown. As the name suggests, online algorithms consider the algorithmic aspects of interactive systems: We wish to design strategies that always compute good output and keep a given system in good state. No assumptions are made about the input stream. The input can even be generated by an adversary that creates new input portions based on the system's reactions to previous ones. We seek algorithms that have a provably good performance. Formally, an online algorithm receives a sequence of requests σ = σ(1), . . . , σ(m). These requests must be served in the order of occurrence. When serving request σ(t), an online algorithm does not know requests σ(t) with t> t. Serving requests incurs cost and the goal is to minimize the total cost paid on the entire request sequence. This process can be viewed as a request answer game. An adversary generates requests and an online algorithm has to serve them one at a time. The performance of online algorithms is usually evaluated using competitive analysis [65]. Here an online algorithm ALG is compared to an optimal offline algorithm OPT that knows the entire request sequence σ in advance and can serve it with minimum cost. Given a sequence σ, let ALG(σ) and OPT(σ) denote the costs incurred by ALG and OPT, respectively. Algorithm ALG is called c-competitive if there exists a constant b such that ALG(σ) ≤ c ' OPT(σ) + b, for all sequences σ. The constant b must be independent of the input σ. We note that competitive analysis is a strong worst-case performance measure.Over the past 15 years online algorithms have received tremendous research interest. Online problems have been studied in many application areas including resource management in operating systems, data structuring, scheduling, networks, and computational finance. In the following sections we first survey fundamental results. We address the paging problem, selforganizing lists, the k-server problem as well as metrical task systems. Then we review a number of new results in application areas of current interest. We focus on algorithmic problems in large networks and competitive auctions. Finally we present refinements of competitive analysis and conclude with some remarks. © 2006 Springer-Verlag Berlin Heidelberg.