Optimal Online Algorithms for File-Bundle Caching and Generalization to Distributed Caching

TL;DR

This paper analyzes optimal online algorithms for file-bundle caching, extending classic cache problems to multiple files per query, and explores distributed caching scenarios with competitive ratio guarantees.

Contribution

It introduces near-optimal online algorithms for file-bundle caching and generalizes the $(h,k)$-paging problem, including distributed cache settings.

Findings

LRU is near-optimal for file-bundle caching.

Provides a randomized $O(l \\ln \\frac{k}{k-h})$-competitive algorithm.

Distributed caching algorithms with competitive ratios of $(l^2+l)$ and $O(l \\ln(2l+1))$.

Abstract

We consider a generalization of the standard cache problem called file-bundle caching, where different queries (tasks), each containing $l\ge 1$ files, sequentially arrive. An online algorithm that does not know the sequence of queries ahead of time must adaptively decide on what files to keep in the cache to incur the minimum number of cache misses. Here a cache miss refers to the case where at least one file in a query is missing among the cache files. In the special case where $l=1$, this problem reduces to the standard cache problem. We first analyze the performance of the classic least recently used (LRU) algorithm in this setting and show that LRU is a near-optimal online deterministic algorithm for file-bundle caching with regard to competitive ratio. We then extend our results to a generalized $(h,k)$-paging problem in this file-bundle setting, where the performance of the online algorithm with a cache size $k$ is compared to an optimal offline benchmark of a smaller cache size $h<k$. In this latter case, we provide a randomized $O(l \ln \frac{k}{k-h})$-competitive algorithm for our generalized $(h,k)$-paging problem, which can be viewed as an extension of the classic marking algorithm. We complete this result by providing a matching lower bound for the competitive ratio, indicating that the performance of this modified marking algorithm is within a factor of two of any randomized online algorithm. Finally, we look at the distributed version of the file-bundle caching problem where there are $m\ge 1$ identical caches in the system. In this case we show that for $m=l+1$ caches, there is a deterministic distributed caching algorithm which is $(l^2+l)$-competitive and a randomized distributed caching algorithm which is $O(l\ln(2l+1))$-competitive when $l\ge 2$.