A note on hardness of diameter approximation

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 4ℓ+2 and 6ℓ+1 requires Ω˜(n) rounds. We slightly tighten this result by showing that even distinguishing between diameter 2ℓ+1 and 3ℓ+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.