AI 中文总结
本文研究内存分配的紧边界，发现采用k=1+o(1)的请求碎片化可将最优竞争比从Θ(log M)降至Θ(log log M)，该结论对确定性和随机算法均成立且为紧界。
AI 中文摘要
经典内存分配问题旨在将不同大小的对象放置在内存中，同时最小化所谓的内存高水位标记。自20世纪70年代初以来，已知任何确定性在线分配器的最优竞争比为Θ(log M)，其中M是底层请求序列的体积高水位标记。本文首先提出一个简单观察：许多现实世界的分配器似乎通过采用略有不同的内存分配模型来规避1971年的下界。这些分配器使用我们所称的k聚合请求碎片化，即只要同时片段的最大数量不超过同时请求的最大数量的k倍，内存分配器就被允许将请求拆分为多个片段。我们考虑以下基本问题：请求碎片化是否从根本上改变了内存分配问题，若是，如何改变？我们的结果带来了几个意外发现。其中，我们发现即使使用k=1+o(1)的请求碎片化，最优竞争比——在经典设置中为Θ(log M)——会降至Θ(log log M)。该结果通过匹配的上界和下界被证明是紧的，适用于确定性和随机算法。
英文摘要
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.
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