In this short note, we give a novel algorithm for $O(1)$ round triangle counting in bounded arboricity graphs. Counting triangles in $O(1)$ rounds (exactly) is listed as one of the interesting remaining open problems in the recent survey of Im et al. [IKLMV23]. The previous paper of Biswas et al. [BELMR20], which achieved the best bounds under this setting, used $O(\log \log n)$ rounds in sublinear space per machine and $O(m\alpha)$ total space where $\alpha$ is the arboricity of the graph and $n$ and $m$ are the number of vertices and edges in the graph, respectively. Our new algorithm is very simple, achieves the optimal $O(1)$ rounds without increasing the space per machine and the total space, and has the potential of being easily implementable in practice.

翻译：在这篇短文中，我们提出了一种新颖的算法，用于在有界树密度图中实现$O(1)$轮次的三角形计数。在$O(1)$轮次中（精确地）计数三角形被列为Im等人近期综述[IKLMV23]中一个有趣且尚未解决的开放问题。此前Biswas等人[BELMR20]的论文在此设定下取得了最佳界限，其使用每台机器次线性空间，总空间为$O(m\alpha)$，需要$O(\log \log n)$轮次，其中$\alpha$为图树密度，$n$和$m$分别为图的顶点数和边数。我们的新算法非常简单，在未增加每台机器空间和总空间的前提下实现了最优的$O(1)$轮次，并具有易于在实践部署的潜力。