Tight Bounds for Memory Allocation With and Without Request Fragmentation

2026-08-28

AI Summary

研究通过引入k-聚合请求碎片化方法，改变了经典内存分配问题中最小化内存高水位线的任务，使得最优竞争比率从Θ(log M)降低到Θ(log log M)。

Abstract

The classical memory-allocation problem captures the task of placing objects of different sizes in memory, while minimizing the so-called memory high-water mark. It has been known since the early 1970s that the optimal competitive ratio for any deterministic online allocator is $Θ(\log M)$, where $M$ is the volume high-water mark of the underlying request sequence. This paper begins with a simple observation: many real-world allocators seem to bypass the 1971 lower bound by adopting a slightly different model for memory allocation. These allocators use what we call $k$-aggregate request fragmentation, meaning that the memory allocator is permitted to break requests into multiple fragments, so long as the all-time maximum number of simultaneous fragments is at most $k$ times the all-time maximum number of simultaneous requests. We consider the following basic question: Does request fragmentation fundamentally change the problem of memory allocation, and if so, how? Our results come with several surprises. Among these, we find that even using $k = 1 + o(1)$ request fragmentation, the optimal competitive ratio---which was $Θ(\log M)$ in the classical setting---collapses to $Θ(\log \log M)$. This result is shown to be tight with matching upper and lower bounds, applying to both deterministic and randomized algorithms.

Problem

• memory allocation
• request fragmentation
• competitive ratio
• deterministic online allocator
• fragmentation model

Innovation

• request fragmentation
• competitive ratio
• memory allocation