Fast distributed algorithms for girth, cycles and small subgraphs

In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain a constant-time algorithm for additive +1 approximation in Congested Clique, and the first parametrized algorithm for exact computation in Congest. In the Congested Clique model, we first develop a technique for learning small neighborhoods, and apply it to obtain an O(1)-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently listing all copies of any subgraph, which is deterministic and improves upon the state-of the-art for non-dense graphs. We give two concrete applications of the partition tree technique: First we show that for constant k, it is possible to solve C_2k-detection in O(1) rounds in the Congested Clique, improving on prior work, which used fast matrix multiplication and thus had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. We remark that no analogous result is currently known for sequential algorithms. In the Congest model, we describe a new approach for finding cycles, and instantiate it in two ways: first, we show a fast parametrized algorithm for girth with round complexity Õ(min{g · n¹−1/Θ(g⁾, n}) for any girth g; and second, we show how to find small even-length cycles C_2k for k = 3, 4, 5 in O(n¹−1^/k) rounds. This is a polynomial improvement upon the previous running times; for example, our C6-detection algorithm runs in O(n²/³) rounds, compared to O(n³/⁴) in prior work. Finally, using our improved C6-freeness algorithm, and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current Ω(∼ ^√n) lower bound for C6-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.