An Optimal Algorithm for Triangle Counting in the Stream

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, ?_E the maximum number of triangles which share a single edge, and ?_V the maximum number of triangles which share a single vertex, then our algorithm requires space: O?(m/T?(?_E + ?{?_V})) Taken with the ?((m ?_E)/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the ?((m ?{?_V})/T) 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