An Optimal Algorithm for Triangle Counting in the Stream

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, DeltaE the maximum number of triangles which share a single edge, and DeltaV the maximum number of triangles which share a single vertex, then our algorithm requires space: [ widetildeOleft(fracmT· left(DeltaE + √DeltaVright)right) ] Taken with the Omegaleft(fracm DeltaETright) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the Omegaleft( fracm √DeltaVTright) 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.