Entropy-based bounds for online algorithms

We focus in this work on an aspect of online computation that is not addressed by standard competitive analysis, namely, identifying request sequences for which nontrivial online algorithms are useful versus request sequences for which all algorithms perform equally poorly. The motivations for this work are advanced system and architecture designs which allow the operating system to dynamically allocate resources to online protocols such as prefetching and caching. To utilize these features, the operating system needs to identify data streams that can benefit from more resources. Our approach in this work is based on the relation between entropy, compression, and gambling, extensively studied in information theory. It has been shown that in some settings, entropy can either fully or at least partially characterize the expected outcome of an iterative gambling game. Our goal is to study the extent to which the entropy of the input characterizes the expected performance of online algorithms for problems that arise in computer applications. We study bounds based on entropy for three classical online problems---list accessing, prefetching, and caching. Our bounds relate the performance of the best online algorithm to the entropy, a parameter intrinsic to characteristics of the request sequence. This is in contrast to the competitive ratio parameter of competitive analysis, which quantifies the performance of the online algorithm with respect to an optimal offline algorithm. For the prefetching problem, we give explicit upper and lower bounds for the performance of the best prefetching algorithm in terms of the entropy of the request sequence. In contrast, we show that the entropy of the request sequence alone does not fully capture the performance of online list accessing and caching algorithms.