分配中的选择-内存权衡
Choice-memory tradeoff in allocations
摘要：
在经典的球和箱范例中，其中$n$球独立且均匀地放置在$n$箱中，通常其中具有至少两个球的箱的数量是$Θ(n)$，并且箱中的球的最大数量是$Θ(\frac{\log n}{\log \log n})$。众所周知，当每一轮为容器提供$k$独立的均匀选项时，当且仅当$k=Ω(\log n)$时，通常可以实现恒定的最大负载。此外，如果使用$k>\log_2n$，则可以避免$n/2$球之间的任何碰撞。在这项工作中，我们将其扩展到只有$m$位内存可用的设置。我们建立了一个权衡的选择$k$和内存$m$，由数量$km/n$。粗略地说，我们表明，对于$km\gg n$，可以实现恒定的最大负载，而对于$km\ll n$，与$k=1$的情况相比，没有实质性的改善（即，随机分配）。对于任何$k=Ω(\log n)$和$m=Ω(\log^2n)$，如果$km=Ω(n)$，则可以实现恒定负载w. h. p.，而如果$km=o(n)$，则负载是无界的。类似地，如果$km>Cn$，则可以在没有任何冲突的情况下分配$n/2$球，而对于$km<εn$，通常存在$Ω(n)$冲突。此外，我们表明，负载是w. h. p.至少$\frac{\log(n/m)}{\log k+\log\log(n/m)}$。特别地，对于$k\leq\operatorname {polylog}(n)$，如果$m=n^{1-δ}$，则最佳最大负载是$Θ(\frac{\log n}{\log\log n})$（与$k=1$的情况相同），而$m=2n$足以确保恒定负载。最后，我们分析了非自适应分配算法，并给出了严格的上限和下限的性能。
英文摘要：
In the classical balls-and-bins paradigm, where $n$ balls are placed independently and uniformly in $n$ bins, typically the number of bins with at least two balls in them is $Θ(n)$ and the maximum number of balls in a bin is $Θ(\frac{\log n}{\log \log n})$. It is well known that when each round offers $k$ independent uniform options for bins, it is possible to typically achieve a constant maximal load if and only if $k=Ω(\log n)$. Moreover, it is possible w.h.p. to avoid any collisions between $n/2$ balls if $k>\log_2n$. In this work, we extend this into the setting where only $m$ bits of memory are available. We establish a tradeoff between the number of choices $k$ and the memory $m$, dictated by the quantity $km/n$. Roughly put, we show that for $km\gg n$ one can achieve a constant maximal load, while for $km\ll n$ no substantial improvement can be gained over the case $k=1$ (i.e., a random allocation). For any $k=Ω(\log n)$ and $m=Ω(\log^2n)$, one can achieve a constant load w.h.p. if $km=Ω(n)$, yet the load is unbounded if $km=o(n)$. Similarly, if $km>Cn$ then $n/2$ balls can be allocated without any collisions w.h.p., whereas for $km<εn$ there are typically $Ω(n)$ collisions. Furthermore, we show that the load is w.h.p. at least $\frac{\log(n/m)}{\log k+\log\log(n/m)}$. In particular, for $k\leq\operatorname {polylog}(n)$, if $m=n^{1-δ}$ the optimal maximal load is $Θ(\frac{\log n}{\log\log n})$ (the same as in the case $k=1$), while $m=2n$ suffices to ensure a constant load. Finally, we analyze nonadaptive allocation algorithms and give tight upper and lower bounds for their performance.
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