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 $\tilde{O}(\min(g\cdot n^{1-1/\Theta(g)},n))$ 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 $\tilde\Omega(\sqrt{n})$ lower bound for $C_6$-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.
翻译:在本文中,我们提供了快速分布的图表算法,用于检测和列出小型子图,以及用于计算或接近 girth。我们的算法通过多元因素改善了艺术状态,对于Girth,我们获得了一个固定时间的算法,用于Congested Clique中的添加+1近似值,以及用于在CONEST中精确计算的第一个简单化算法。在Congest Clique中,我们开发了一个学习小区的技术,并应用它来获得一个只用添加+1错误来计算 girth 的 Ol($) 。我们引入了一种新的技术(分区树技术),以便有效和确定地列出任何子图的复制件。我们给出了这种技术的两个应用:首先,对于常数美元, $C\2k} 和oktrequestal4, 在Congrique 中可以找到一个小数回合中的美元, 美元, 用来显示一个最接近的变数的方法。