围长、圈和小子图的快速分布式算法
Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs
摘要：
在本文中，我们给快速分布式图算法检测和上市的小子图，并计算或近似的围长。我们的算法通过多项式因子改进了现有技术，对于围长，我们得到了拥塞集团中加性+1近似的常数时间算法，以及CONGEST中精确计算的第一个参数化算法。在拥挤的集团，我们开发了一种技术，学习小的邻域，并将其应用到获得一个$O(1)$轮算法，计算周长只有一个加性+1错误。接下来，我们介绍了一种新的技术（分区树技术），允许有效地和确定性地列出任何子图的所有副本，改进了非稠密图的最新技术。我们给出了这种技术的两个应用：首先，我们表明，对于恒定的$k$，$C_{2k}$ -检测可以解决在$O(1)$轮拥塞集团，改善以前的工作，使用矩阵乘法和多项式轮复杂度。第二，我们证明了在无三角形图中，围长可以在时间上以多项式的速度精确计算，比一般图的最佳已知界要快。在CONGEST中，我们描述了一种新的方法来寻找周期，并将其应用在两个方面：首先，我们展示了一个快速的参数化算法周长与轮复杂度$\tilde{O}(\min(g\cdot n^{1-1/Θ(g)},n))$的任何周长$g$;第二，我们展示了如何找到小的偶数长度周期$C_{2k}$的$k = 3,4,5$在$O(n^{1-1/k})$轮，这是一个多项式的改善以前的运行时间。最后，利用我们改进的$C_6$ -freeness算法和Eden等人关于证明三角形-freeness下界的障碍，我们表明，通过任何多项式因子改进Korhonen等人的$C_6$ -freeness的当前$\tildeΩ(\sqrt{n})$下限将意味着强电路复杂性下限。
英文摘要：
In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently and deterministically listing all copies of any subgraph, improving upon the state-of the-art for non-dense graphs. We give two applications of this technique: First we show that for constant $k$, $C_{2k}$-detection can be solved in $O(1)$ rounds in the Congested Clique, improving on prior work which used matrix multiplication and had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. In CONGEST, we describe a new approach for finding cycles, and apply it in two ways: first we show a fast parametrized algorithm for girth with round complexity $\tilde{O}(\min(g\cdot n^{1-1/Θ(g)},n))$ for any girth $g$; and second, we show how to find small even-length cycles $C_{2k}$ for $k = 3,4,5$ in $O(n^{1-1/k})$ rounds, which is a polynomial improvement upon the previous running times. Finally, using our improved $C_6$-freeness algorithm and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current $\tildeΩ(\sqrt{n})$ lower bound for $C_6$-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
