An Optimal Algorithm for Triangle Counting in the Stream

TL;DR

This paper introduces an optimal streaming algorithm for approximating triangle counts in graphs, matching known lower bounds and resolving a longstanding problem in graph streaming complexity.

Contribution

The paper presents a new space-efficient algorithm for triangle counting in streams that is proven to be optimal up to logarithmic factors, based on matching lower bounds.

Findings

The algorithm achieves space complexity close to the theoretical lower bounds.

It is optimal up to log factors, resolving the complexity of triangle counting in streaming models.

The approach improves understanding of the fundamental limits in graph streaming algorithms.

Abstract

We present a new algorithm for approximating the number of triangles in a graph $G$ whose edges arrive as an arbitrary order stream. If $m$ is the number of edges in $G$, $T$ the number of triangles, $\Delta_E$ the maximum number of triangles which share a single edge, and $\Delta_V$ the maximum number of triangles which share a single vertex, then our algorithm requires space: \[ \widetilde{O}\left(\frac{m}{T}\cdot \left(\Delta_E + \sqrt{\Delta_V}\right)\right) \] Taken with the $\Omega\left(\frac{m \Delta_E}{T}\right)$ lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the $\Omega\left( \frac{m \sqrt{\Delta_V}}{T}\right)$ lower bound of Kallaugher and Price (SODA 2017), our algorithm is optimal up to log factors, resolving the complexity of a classic problem in graph streaming.