Improved Massively Parallel Triangle Counting in $O(1)$ Rounds
Quanquan C. Liu, C. Seshadhri
TL;DR
The paper addresses exact triangle counting in the MPC model for graphs with bounded arboricity $\alpha$ and achieves a constant-round algorithm. It leverages wedge enumeration anchored at the endpoint with smaller degree, partitioning adjacencies into $n^{\delta}$-sized chunks and using MPC primitives to form and validate wedges, with a total space of $O(m\alpha)$ and per-machine space $O(n^{\delta})$, requiring $O(1/\delta)$ rounds which is constant for fixed $\delta$. A matching lower bound of $\Omega(1/\delta)$ rounds under the same space constraints demonstrates near-tightness. The work yields a simple, practical, and scalable approach for exact triangle counting in real-world graphs where arboricity is small, with potential impact on large-scale graph analytics and database joins.
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 $O(mα)$ total space where $α$ 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.