A Note on Hardness of Diameter Approximation

We revisit the hardness of approximating the diameter of a network. In the CONGEST model, $\tilde \Omega (n)$ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12]. Abboud et al. DISC 2016 extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer $1 \leq \ell \leq \operatorname{polylog} (n)$, distinguishing between networks of diameter $4 \ell + 2$ and $6 \ell + 1$ requires $\tilde \Omega (n)$ rounds. We slightly tighten this result by showing that even distinguishing between diameter $2 \ell + 1$ and $3 \ell + 1$ requires $\tilde \Omega (n)$ 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. This is suited for teaching both the lower bound in the CONGEST model and the conditional lower bound in the RAM model.