How the Degeneracy Helps for Triangle Counting in Graph Streams
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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±ε)-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 Θ(min(m^3/2/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, κ, 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 O(mκ/T). For constant degeneracy graphs, this bound is O(m/T), which is significantly smaller than both m^3/2/T and m/√(T). We complement our algorithmic result with a nearly matching lower bound of Ω(mκ/T).
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