An Optimal Algorithm for Triangle Counting

05/04/2021

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by   Rajesh Jayaram, et al.

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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, Δ_E the maximum number of triangles which share a single edge, and Δ_V the maximum number of triangles which share a single vertex, then our algorithm requires space: O(m/T·(Δ_E + √(Δ_V))) Taken with the Ω(m Δ_E/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the Ω( m √(Δ_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.