并行排序中划分的最优轮数与样本量复杂度
Optimal Round and Sample-Size Complexity for Partitioning in Parallel Sorting
AI总结：
本文研究并行排序中均衡划分的采样与直方图统计轮数及样本量复杂度，证明$O(\log^* p)$轮且每轮$O(p/\log^* p)$样本的上界及匹配下界，并证明一轮算法的最优样本下界为$\Omega(p \log p)$。
摘要：
面向分布式内存架构的最先进并行排序算法基于通过采样和直方图统计来计算均衡划分。通过找到将已排序键划分为大小均匀块的样本，这些算法最小化了所需的通信轮数。直方图统计（计算样本位置）指导采样，使得所收集样本的总数得以减少。我们推导了计算均衡划分所需采样/直方图统计轮数的下界和上界。我们改进了先前的结果，证明当使用$p$个处理器时，$O(\log^* p)$轮且每轮$O(p/\log^* p)$个样本即已足够。我们以一个下界与之匹配，该下界表明任何每轮使用$O(p)$个样本的算法至少需要$\Omega(\log^* p)$轮。此外，我们证明了一轮情况下的$\Omega(p \log p)$样本下界，从而证明现有的一轮算法：sample sort、AMS sort和HSS具有最优的样本量复杂度。为推导该下界，我们提出了一种困难的随机输入分布，并应用了游程分布理论中的经典结果。
英文摘要：
State-of-the-art parallel sorting algorithms for distributed-memory architectures are based on computing a balanced partitioning via sampling and histogramming. By finding samples that partition the sorted keys into evenly-sized chunks, these algorithms minimize the number of communication rounds required. Histogramming (computing positions of samples) guides sampling, enabling a decrease in the overall number of samples collected. We derive lower and upper bounds on the number of sampling/histogramming rounds required to compute a balanced partitioning. We improve on prior results to demonstrate that when using $p$ processors, $O(\log^* p)$ rounds with $O(p/\log^* p)$ samples per round suffice. We match that with a lower bound that shows that any algorithm with $O(p)$ samples per round requires at least $Ω(\log^* p)$ rounds. Additionally, we prove the $Ω(p \log p)$ samples lower bound for one round, thus proving that existing one round algorithms: sample sort, AMS sort and HSS have optimal sample size complexity. To derive the lower bound, we propose a hard randomized input distribution and apply classical results from the distribution theory of runs.
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