A note on hardness of diameter approximation

Abstract We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, Ω ˜ ( n ) rounds are necessary to compute the diameter (Frischknecht et al., 2012 [2] ), where Ω ˜ ( ⋅ ) hides polylogarithmic factors. Abboud et al. (2016) [3] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer 1 ≤ l ≤ polylog ( n ) , distinguishing between networks of diameter 4 l + 2 and 6 l + 1 requires Ω ˜ ( n ) rounds. We slightly tighten this result by showing that even distinguishing between diameter 2 l + 1 and 3 l + 1 requires Ω ˜ ( n ) rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition.