Extending the accommodating function

Abstract.

The applicability of the accommodating function, a relatively new measure for the quality of on-line algorithms, is extended.

The standard quality measure for on-line algorithms is the competitive ratio, which is, roughly speaking, the worst case ratio of the on-line performance to the optimal off-line performance. However, for many on-line problems, the competitive ratio gives overly pessimistic results and/or fails to distinguish between algorithms that are known to perform very differently in practice. Many researchers have proposed variations on the competitive ratio to obtain more realistic results. These variations are often tailor-made for specific on-line problems.

The concept of the accommodating function applies to any on-line problem with some limited resource, such as bins, seats in a train, or pages in a cache. If a limited amount n of some resource is available, the accommodating function $\mathcal{A}(\alpha)$ is the competitive ratio when input sequences are restricted to those for which the amount $\alpha n$ of resources suffices for an optimal off-line algorithm. For all resource bounded problems, the standard competitive ratio is $\lim_{\alpha\rightarrow\infty}\mathcal{A}(\alpha)$.

The accommodating function was originally used only for $\alpha\geq 1$. We focus on $\alpha < 1$, observe that the function now appears interesting for a greater variety of problems, and use it to make new distinctions between known algorithms and to find new ones.