论文标题
在原木直径时间中的婴儿车的连接组件
Connected Components on a PRAM in Log Diameter Time
论文作者
论文摘要
我们提出了一个$ O（\ log D + \ log \ log_ {m/n} n）$ - 时间随机的婴儿车算法，用于计算$ n $ vertex的连接组件，$ m $ - $ m $ - 无方向的图形，带有最大组件二十级$ d $。该算法使用$ o（m）$处理器在任意CRCW（并发，并发写入与任意写入分辨率）的情况下运行。时间界的可能性很大。 我们的算法基于Andoni等人的突破性结果。 [FOCS'18]和Behnezhad等。 [焦点19]。他们的算法在更强大的MPC模型上运行，并依赖于$ o（1）$时间的排序和计算前缀总和，在带有$ \ text {poly}（poly}（n）$ processors的CRCW PRAM上，在CRCW PRAM上服用$ω（\ log N / \ log \ log \ log n）$ time的任务。我们的简单算法使用有限的散布哈希，不排序或进行前缀总和。它与Behnezhad等人的算法的时间和空间界限相匹配，后者改善了Andoni等人的时间。 人们普遍认为，每个处理器和MPC模型的局部计算较大的私人内存允许算法比婴儿车更快。我们的结果表明，至少对于基本的图形问题，例如连接的组件和跨越森林，可能是不需要的。
We present an $O(\log d + \log\log_{m/n} n)$-time randomized PRAM algorithm for computing the connected components of an $n$-vertex, $m$-edge undirected graph with maximum component diameter $d$. The algorithm runs on an ARBITRARY CRCW (concurrent-read, concurrent-write with arbitrary write resolution) PRAM using $O(m)$ processors. The time bound holds with good probability. Our algorithm is based on the breakthrough results of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. Their algorithms run on the more powerful MPC model and rely on sorting and computing prefix sums in $O(1)$ time, tasks that take $Ω(\log n / \log\log n)$ time on a CRCW PRAM with $\text{poly}(n)$ processors. Our simpler algorithm uses limited-collision hashing and does not sort or do prefix sums. It matches the time and space bounds of the algorithm of Behnezhad et al., who improved the time bound of Andoni et al. It is widely believed that the larger private memory per processor and unbounded local computation of the MPC model admit algorithms faster than that on a PRAM. Our result suggests that such additional power might not be necessary, at least for fundamental graph problems like connected components and spanning forest.
