# A Note on Hardness of Diameter Approximation

## TL;DR

This paper refines the understanding of the distributed complexity of approximating network diameter, establishing tighter lower bounds and clarifying the connection to the orthogonal vectors problem.

## Contribution

It tightens existing lower bounds for diameter approximation in distributed networks and explicitly links these bounds to the orthogonal vectors problem.

## Key findings

- Distinguishing diameters 2l+1 and 3l+1 requires rac{{n}}{{polylog(n)}} rounds.
- Lower bounds apply to sparse graphs and specific diameter ranges.
- Connection to orthogonal vectors problem simplifies the conceptual framework.

## Abstract

We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, $ \tilde \Omega (n) $ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12], where $ \tilde \Omega (\cdot) $ hides polylogarithmic factors. 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.