How the Degeneracy Helps for Triangle Counting in Graph Streams

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±varepsilon)-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(m3/2/T, m/√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 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 widetildeO(mkappa/T). For constant degeneracy graphs, this bound is widetildeO(m/T), which is significantly smaller than both m3/2/T and m/√T. We complement our algorithmic result with a nearly matching lower bound of Omega(mkappa/T).