Near-linear lower bounds for distributed distance computations, even in sparse networks

We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an Ω(n) lower bound for computing the diameter in sparse networks, which was previously known only for dense networks. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show Ω(n) lower bounds for computing (formula presented)-approximations of the diameter or the radius, and for computing a (formula presented)-approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors, and for eccentricities the improvement is both in the lower bound and in the approximation factor. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of (α, β)-spanners, for α < β + 1.