Memory Reallocation with Polylogarithmic Overhead

TL;DR

This paper introduces a memory allocator with polylogarithmic expected overhead for online memory reallocation, significantly improving previous bounds and approaching theoretical lower limits.

Contribution

It presents a new online memory reallocation algorithm with polylogarithmic expected overhead, utilizing the sunflower lemma, and proves that high-probability subpolynomial overhead is impossible.

Findings

Achieves worst-case expected overhead of polylogarithmic order.

Improves previous overhead bounds exponentially.

Proves the necessity of expected-overhead guarantees due to inherent limitations.

Abstract

The Memory Reallocation problem asks to dynamically maintain an assignment of given objects of various sizes to non-overlapping contiguous chunks of memory, while supporting updates (insertions/deletions) in an online fashion. The total size of live objects at any time is guaranteed to be at most a $1-\epsilon$ fraction of the total memory. To handle an online update, the allocator may rearrange the objects in memory to make space, and the overhead for this update is defined as the total size of moved objects divided by the size of the object being inserted/deleted. Our main result is an allocator with worst-case expected overhead $\mathrm{polylog}(\epsilon^{-1})$. This exponentially improves the previous worst-case expected overhead $\tilde O({\epsilon}^{-1/2})$ achieved by Farach-Colton, Kuszmaul, Sheffield, and Westover (2024), narrowing the gap towards the $\Omega(\log\epsilon^{-1})$ lower bound. Our improvement is based on an application of the sunflower lemma previously used by Erd\H{o}s and S\'{a}rk\"{o}zy (1992) in the context of subset sums. Our allocator achieves polylogarithmic overhead only in expectation, and sometimes performs expensive rebuilds. Our second technical result shows that this is necessary: it is impossible to achieve subpolynomial overhead with high probability.