How the Degeneracy Helps for Triangle Counting in Graph Streams

Abstract

We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of $m$ edges, our aim is to compute a $(1\pm \epsilon )$-approximation to the triangle count $T$, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially $\Theta (\min (m^{3/2}/T,m/\sqrt{T}))$ (McGregor et al., PODS 2016, Bera et al., STACS 2017). We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for \emph{low degeneracy graphs}. The degeneracy, $\kappa$, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity $\tilde{O}(m\kappa /T)$. For constant degeneracy graphs, this bound is $\tilde{O}(m/T)$, which is significantly smaller than both $m^{3/2}/T$ and $m/\sqrt{T}$. We complement our algorithmic result with a nearly matching lower bound of $\Omega (m\kappa /T)$.

Accepted for publication in PODS'2020