There is a trivial algorithm for approximatetriangle counting where $T$ is the number of triangles in the graph and $n$ thenumber of vertices . At the same time, one may count triangles exactly usingfast matrix multiplication in time $O . We answer this question positively by providing an algorithm which runs in time . This is optimal in the sense that as long as the exponent of $T is independent of $n, T$, it cannot be improvedwhile retaining the dependency on $n . We also consider the problem of approximate triangle counting in sparsegraphs, parameterizing by number of edges $m$. The best known algorithmruns in time $\tilde{O\big(\frac{m^{3/2)$ [Eden et al., SIAM Journalon Computing, 2017]. There is also also a well known algorithm for exact trianglecounting that runs in $tribal counting that run in time. We again get an algorithm that runs . This algorithm that retains the exponent $m

**Author(s) :**Jakub TÄ›tek

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Keywords : time - algorithm - counting - runs - approximate -