# Triangle Finding and Listing in CONGEST Networks

## TL;DR

This paper introduces the first sublinear round complexity algorithms for triangle detection and listing in the standard CONGEST network model, advancing distributed graph algorithms.

## Contribution

It presents the first sublinear algorithms for triangle finding and listing in the CONGEST model, with specific complexities and a matching lower bound.

## Key findings

- Triangle finding algorithm with $O(n^{2/3}( ext{log} n)^{2/3})$ rounds.
- Triangle listing algorithm with $O(n^{3/4} ext{log} n)$ rounds.
- Lower bound of $ ilde{ ext{Omega}}(n^{1/3})$ rounds for triangle listing.

## Abstract

Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology of the network. More precisely, we give randomized algorithms for triangle finding and listing with round complexity $O(n^{2/3}(\log n)^{2/3})$ and $O(n^{3/4}\log n)$, respectively, where $n$ denotes the number of nodes of the network. We also show a lower bound $\Omega(n^{1/3}/\log n)$ on the round complexity of triangle listing, which also holds for the CONGEST clique model.