Improved Massively Parallel Triangle Counting in $O(1)$ Rounds

TL;DR

This paper presents a simple, optimal $O(1)$ round algorithm for triangle counting in bounded arboricity graphs, solving a notable open problem with practical potential.

Contribution

The authors introduce a novel $O(1)$ round algorithm for triangle counting that matches the best bounds without increasing space, improving upon prior work.

Findings

Achieves $O(1)$ rounds for triangle counting in bounded arboricity graphs.

Uses the same space complexity as previous best algorithms.

Potential for practical implementation and application.

Abstract

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.