Non-Linear Paging

TL;DR

This paper introduces non-linear paging, a general model encompassing weighted, submodular, and supermodular paging, and develops tight algorithms and bounds for this broad setting, advancing understanding of cache management.

Contribution

It formulates a new non-linear paging model, introduces a novel parameter and LP approach, and provides tight algorithms and bounds for general and supermodular paging.

Findings

Developed a tight deterministic ompetitive algorithm for non-linear paging.

Established a lower bound of o(al log^2 (ll)) for randomized algorithms.

Provided polylogarithmic bounds and offline algorithms for supermodular paging.

Abstract

We formulate and study non-linear paging - a broad model of online paging where the size of subsets of pages is determined by a monotone non-linear set function of the pages. This model captures the well-studied classic weighted paging and generalized paging problems, and also submodular and supermodular paging, studied here for the first time, that have a range of applications from virtual memory to machine learning. Unlike classic paging, the cache threshold parameter $k$ does not yield good competitive ratios for non-linear paging. Instead, we introduce a novel parameter $\ell$ that generalizes the notion of cache size to the non-linear setting. We obtain a tight deterministic $\ell$-competitive algorithm for general non-linear paging and a $o\left(\log^2 (\ell)\right)$-competitive lower bound for randomized algorithms. Our algorithm is based on a new generic LP for the problem that captures both submodular and supermodular paging, in contrast to LPs used for submodular cover settings. We finally focus on the supermodular paging problem, which is a variant of online set cover and online submodular cover, where sets are repeatedly requested to be removed from the cover. We obtain polylogarithmic lower and upper bounds and an offline approximation algorithm.