A Nearly Quadratic Improvement for Memory Reallocation

TL;DR

This paper presents a nearly quadratic improvement in the expected update cost for the Memory Reallocation Problem, surpassing previous bounds and disproving conjectures about optimality for larger items.

Contribution

It introduces an allocator with expected update cost $O(rac{1}{ oot 2 ext{ of } ext{epsilon}} ext{ polylog} ext{ of } rac{1}{ ext{epsilon}})$ for all input sequences, and provides the first non-trivial lower bound for the problem.

Findings

Achieves $O(rac{1}{ oot 2 ext{ of } ext{epsilon}} ext{ polylog} ext{ of } rac{1}{ ext{epsilon}})$ expected update cost.

Proves a lower bound of $oldsymbol{ ext{Omega}( ext{log} rac{1}{ ext{epsilon}})}$ for any allocator.

Shows that in stochastic sequences with random sizes, $O( ext{log} rac{1}{ ext{epsilon}})$ cost is achievable.

Abstract

In the Memory Reallocation Problem a set of items of various sizes must be dynamically assigned to non-overlapping contiguous chunks of memory. It is guaranteed that the sum of the sizes of all items present at any time is at most a $(1-\varepsilon)$-fraction of the total size of memory (i.e., the load-factor is at most $1-\varepsilon$). The allocator receives insert and delete requests online, and can re-arrange existing items to handle the requests, but at a reallocation cost defined to be the sum of the sizes of items moved divided by the size of the item being inserted/deleted. The folklore algorithm for Memory Reallocation achieves a cost of $O(\varepsilon^{-1})$ per update. In recent work at FOCS'23, Kuszmaul showed that, in the special case where each item is promised to be smaller than an $\varepsilon^4$-fraction of memory, it is possible to achieve expected update cost $O(\log\varepsilon^{-1})$. Kuszmaul conjectures, however, that for larger items the folklore algorithm is optimal. In this work we disprove Kuszmaul's conjecture, giving an allocator that achieves expected update cost $O(\varepsilon^{-1/2} \operatorname*{polylog} \varepsilon^{-1})$ on any input sequence. We also give the first non-trivial lower bound for the Memory Reallocation Problem: we demonstrate an input sequence on which any resizable allocator (even offline) must incur amortized update cost at least $\Omega(\log\varepsilon^{-1})$. Finally, we analyze the Memory Reallocation Problem on a stochastic sequence of inserts and deletes, with random sizes in $[\delta, 2 \delta]$ for some $\delta$. We show that, in this simplified setting, it is possible to achieve $O(\log\varepsilon^{-1})$ expected update cost, even in the ``large item'' parameter regime ($\delta > \varepsilon^4$).