Engineering Nearly Linear-time Algorithms for Small Vertex Connectivity

Q2 Mathematics Journal of Experimental Algorithmics Pub Date : 2021-03-29 DOI:10.1145/3564822
Max Franck, Sorrachai Yingchareonthawornchai
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Abstract

Vertex connectivity is a well-studied concept in graph theory with numerous applications. A graph is k-connected if it remains connected after removing any k −1 vertices. The vertex connectivity of a graph is the maximum k such that the graph is k-connected. There is a long history of algorithmic development for efficiently computing vertex connectivity. Recently, two near linear-time algorithms for small k were introduced by Forster et al. [SODA 2020]. Prior to that, the best-known algorithm was one by Henzinger et al. [FOCS 1996] with quadratic running time when k is small. In this article, we study the practical performance of the algorithms by Forster et al. In addition, we introduce a new heuristic on a key subroutine called local cut detection, which we call degree counting. We prove that the new heuristic improves space-efficiency (which can be good for caching purposes) and allows the subroutine to terminate earlier. According to experimental results on random graphs with planted vertex cuts, random hyperbolic graphs, and real-world graphs with vertex connectivity between 4 and 8, the degree counting heuristic offers a factor of 2–4 speedup over the original non-degree counting version for small graphs and almost 20 times for some graphs with millions of edges. It also outperforms the previous state-of-the-art algorithm by Henzinger et al., even on relatively small graphs.
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小顶点连通性的近似线性时间算法工程
顶点连通性是图论中一个被广泛研究的概念,有着广泛的应用。如果一个图在去掉任意k−1个顶点后仍然保持连通,那么这个图就是k连通的。图的顶点连通性是使图是k连通的最大k。高效计算顶点连通性的算法已经有了很长的发展历史。最近,Forster等人[SODA 2020]引入了两种小k的近线性时间算法。在此之前,最著名的算法是Henzinger等人[FOCS 1996]的算法,当k很小时,它的运行时间是二次的。在本文中,我们研究了Forster等人的算法的实际性能。此外,我们在关键子程序中引入了一种新的启发式方法,称为局部切割检测,我们称之为度计数。我们证明了新的启发式方法提高了空间效率(这对于缓存目的很有好处),并允许子例程提前终止。实验结果表明,在具有种植顶点切割的随机图、随机双曲图和顶点连接在4 ~ 8之间的现实图中,对于小图,度计数启发式算法的速度比原始的非度计数版本提高了2 ~ 4倍,对于具有数百万条边的图,度计数启发式算法的速度提高了近20倍。即使在相对较小的图上,它也优于Henzinger等人先前最先进的算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Experimental Algorithmics
Journal of Experimental Algorithmics Mathematics-Theoretical Computer Science
CiteScore
3.10
自引率
0.00%
发文量
29
期刊介绍: The ACM JEA is a high-quality, refereed, archival journal devoted to the study of discrete algorithms and data structures through a combination of experimentation and classical analysis and design techniques. It focuses on the following areas in algorithms and data structures: ■combinatorial optimization ■computational biology ■computational geometry ■graph manipulation ■graphics ■heuristics ■network design ■parallel processing ■routing and scheduling ■searching and sorting ■VLSI design
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