The Complexity of Counting Cycles in the Adjacency List Streaming Model

We study the problem of counting cycles in the adjacency list streaming model, fully resolving in which settings there exist sublinear space algorithms. Our main upper bound is a two-pass algorithm for estimating triangles that uses $\wtO (m/T^2/3 )$ space, where m is the edge count and T is the triangle count of the graph. On the other hand, we show that no sublinear space multipass algorithm exists for counting $\ell$-cycles for $\ell \geq 5$. Finally, we show that counting 4-cycles is intermediate: sublinear space algorithms exist in multipass but not single-pass settings.