A second look at counting triangles in graph streams

In this paper we present improved results on the problem of counting triangles in edge streamed graphs. For graphs with m edges and at least T   triangles, we show that an extra look over the stream yields a two-pass streaming algorithm that uses O(mϵ4.5T) space and outputs a (1+ϵ)(1+ϵ) approximation of the number of triangles in the graph. This improves upon the two-pass streaming tester of Braverman et al. [2], which distinguishes between triangle-free graphs and graphs with at least T   triangles using O(mT1/3) space. Also, in terms of dependence on T, we show that more passes would not lead to a better space bound. In other words, we prove there is no constant pass streaming algorithm that distinguishes between triangle-free graphs from graphs with at least T   triangles using O(mT1/2+ρ) space for any constant ρ≥0ρ≥0.