多字单写者寄存器空间复杂度的上界与下界
Upper and Lower Bounds on the Space Complexity of Multi-word Single-Writer Registers
浏览论文内容
中文总结 AI 辅助
该研究针对单写者多字寄存器，证明了其空间复杂度的匹配上下界，提出了渐近紧的无等待模拟算法，改进了此前的空间上界，提升了多字寄存器模拟的空间效率。
中文摘要 AI 辅助
我们证明了用较小的共享寄存器模拟大型共享寄存器的空间复杂度的匹配上界与下界。我们聚焦于模拟的寄存器和基础寄存器均为单写者的情况，即它们可被多个读者并发访问，但仅能被单个写者访问。为强化下界，我们证明即使基础寄存器是原子的且模拟的寄存器是规则的，这些下界依然成立。此外，这些下界适用于无阻碍（obstruction-free）实现，因此也适用于无锁（lock-free）和无等待（wait-free）实现。若m为大型寄存器可表示的值的数量，b为每个基础寄存器可表示的值的数量，我们的第一个下界表明：任何存在不可见读者（invisible reader，即从不向基础寄存器写入的读者）的无阻碍实现，至少需要⌈(m-1)/(b-1)⌉个基础寄存器。该下界对不可见读者情形是渐近紧的，且比此前已知的最优下界有指数级提升。对于允许任意可见与不可见读者组合的一般情形，我们证明了一个⌈min((m-1)/(b-1), r + log m / log b)⌉的空间下界，其中r为读者数量。为证明该下界是渐近紧的，我们开发了一种无等待算法，用Θ(r + log m / log b)的空间从原子基础寄存器模拟多字原子寄存器。将该算法与已知的不可见读者构造相结合，得到了一个Θ(min(m/b, r + log m / log b))的空间上界，这改进了此前已知的Θ(min(m/b, r·log m / log b))的空间上界。
英文摘要
We prove matching upper and lower bounds on the space complexity of simulating a large shared register using smaller shared registers. We focus on the case where both the simulated and base registers are single-writer, which means they can be accessed concurrently by multiple readers but only by a single writer. To strengthen our lower bounds, we prove that they hold even when the base registers are atomic and the simulated register is regular. Furthermore, the lower bounds hold for obstruction-free implementations, which means they also hold for lock-free and wait-free implementations. If $m$ is the number of values representable by the large register and $b$ is the number of values representable by each base register, our first lower bound says that any obstruction-free implementation that has an invisible reader requires at least $\lceil \frac{m-1}{b-1} \rceil$ base registers. A reader is considered invisible if it never writes to base registers. This lower bound is asymptotically tight for the invisible-reader case and represents an exponential improvement over the previous best known lower bound. For the general case, which allows any combination of visible and invisible readers, we prove a $\lceil \min(\frac{m-1}{b-1}, r+\frac{\log{m}}{\log{b}}) \rceil$ space lower bound, where $r$ is the number of readers. To show that this lower bound is asymptotically tight, we develop a wait-free algorithm for simulating a multi-word atomic register from atomic base registers using $Θ(r + \frac{\log{m}}{\log{b}})$ space. Combining this algorithm with known invisible-reader constructions gives a $Θ(\min(\frac{m}{b}, r + \frac{\log{m}}{\log{b}}))$ space upper bound. This improves upon the previously known space upper bound of $Θ(\min(\frac{m}{b}, r\frac{\log{m}}{\log{b}}))$.
