Introduction

In this chapter, we consider a very general and abstract online problem that generalizes various problems already studied in this book, e.g., paging and list accessing. For any generic problem, the cost paid by any online algorithm on the arrival of a new request is a function of its current state and the action taken to fulfil the new request. Typically, the chosen action also alters the state of the algorithm, which then determines the subsequent costs. To model this interplay between cost and state transitions, an abstract paradigm called the metrical task system (MTS) is defined, where there is a set of all possible states (one of them is occupied by any online algorithm at any time).

Requests arrive over time, and the cost of an online algorithm to serve or fulfil each request depends on the state from which it chooses to fulfil the request. This cost is called the state dependent cost. If the online algorithm serves the newly arrived request from its current state, then the only cost it pays is the state dependent cost. Otherwise, it first transitions to a new state and then serves the request. In the event of a transition, the algorithm pays the state dependent cost of the new state in addition to the switching cost to move from the present state to the new state. The switching cost is restricted to satisfying the usual metric properties, e.g., the triangle inequality. The overall cost of an online algorithm is the sum of the state dependent cost and the switching cost, summed over all requests. This is a very generic formulation and can model any finite state dependent dynamical system.

We begin this chapter by deriving a lower bound on the competitive ratio of any deterministic algorithm for the MTS. Because of the generality of the MTS, the power of deterministic online algorithms is limited, and the competitive ratio of any deterministic online algorithm is at least 2|S| − 1, where |S| is the total number of states. Thus, more the number of states, more is the power that adversary has over an online algorithm. We next present a simple algorithm called work-function, based on the broad principle of dynamic programming, that achieves this lower bound exactly.