A Nearly Quadratic Improvement for Memory Reallocation
Martin Farach-Colton, William Kuszmaul, Nathan Sheffield, Alek Westover
TL;DR
The paper addresses the Memory Reallocation problem under a load factor of at most $1-\varepsilon$, seeking to minimize online reallocation cost. It introduces new allocators that beat the folklore $O(\varepsilon^{-1})$ bound for arbitrary item sizes, achieving $O(\varepsilon^{-1/2}\operatorname{polylog} \varepsilon^{-1})$ in expectation, and proves a non-trivial lower bound of $\Omega(\log \varepsilon^{-1})$ on update cost for certain input sequences. It also analyzes a stochastic setting with random item sizes, obtaining $O(\log \varepsilon^{-1})$ expected updates, and develops a suite of techniques (SIMPLE, GEO, FLEXHASH, TINYHASH, and RSUM) including nested responsibility suffixes and geometric size classes to organize memory efficiently. Overall, the results substantially improve prior bounds and illuminate a potential structure-randomness trade-off in memory reallocation problems, with practical implications for dynamic storage allocators.
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 $Ω(\log\varepsilon^{-1})$. Finally, we analyze the Memory Reallocation Problem on a stochastic sequence of inserts and deletes, with random sizes in $[δ, 2 δ]$ for some $δ$. 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 ($δ> \varepsilon^4$).