arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1909.00844cs.DScs.DC
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基于随机$2$ -Out收缩的边连通性快速算法

Faster Algorithms for Edge Connectivity via Random $2$-Out Contractions

Mohsen Ghaffari, Krzysztof Nowicki, Mikkel Thorup

备注：
algorithms and data structures, graph algorithms, edge connectivity, out-contractions, randomized algorithms, distributed algorithms, massively parallel computation
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摘要：

我们提供了一个简单的新的随机收缩的方法来解决简单无向图的全局最小割问题。收缩利用来自每个顶点的2-out边缘采样，而不是标准的均匀边缘采样。我们展示了我们的新方法的力量，通过获得更好的算法的顺序，分布式和并行计算模型。我们的最终结果包括以下随机算法计算边缘连接的高概率：-两个顺序算法的复杂性$O(m \log n)$和$O(m+n \log^3 n)$。这些改进了一长串的发展，包括Karger的著名$O(m \log^3 n)$算法[STOC'96]和Henzinger等人的最新$O(m \log^2 n (\log\log n)^2)$算法[SODA'17]。此外，我们的$O(m+n \log^3 n)$算法是最佳的，只要$m = Ω(n \log^3 n)$。在我们新的时间界限whp内，我们还可以构造所有最小割的仙人掌表示。--$Õ(n^{0.8} D^{0.2} + n^{0.9})$轮分布式算法，其中D表示图直径。这大大改进了Daga等人[STOC'19]的最近突破，其实现了$Õ(n^{1-1/353}D^{1/353} + n^{1-1/706})$的轮复杂度，因此提供了用于精确计算边缘连通性的第一个次线性分布式算法。--第一个$O(1)$轮算法，用于大规模并行计算设置，每台机器具有线性内存。

英文摘要：

We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity with high probability: -- Two sequential algorithms with complexities $O(m \log n)$ and $O(m+n \log^3 n)$. These improve on a long line of developments including a celebrated $O(m \log^3 n)$ algorithm of Karger [STOC'96] and the state of the art $O(m \log^2 n (\log\log n)^2)$ algorithm of Henzinger et al. [SODA'17]. Moreover, our $O(m+n \log^3 n)$ algorithm is optimal whenever $m = Ω(n \log^3 n)$. Within our new time bounds, whp, we can also construct the cactus representation of all minimal cuts. -- An $Õ(n^{0.8} D^{0.2} + n^{0.9})$ round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity of $Õ(n^{1-1/353}D^{1/353} + n^{1-1/706})$, hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. -- The first $O(1)$ round algorithm for the massively parallel computation setting with linear memory per machine.

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