G. Cormode and H. Jowhari. A second look at counting triangles in graph streams. Theoretical Computer Science, 552:44-51, 2014.

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((m)/(ε4.5sqrt(T))) space and outputs a (1+ε) approximation of the number of triangles in the graph. This improves upon the two-pass streaming tester of Braverman, Ostrovsky and Vilenchik, ICALP 2013, which distinguishes between triangle-free graphs and graphs with at least T triangle using O((m)/(T1/3)) 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((m)/(T1/2+ρ)) space for any constant ρ>=0.

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