Deterministic distributed construction of linear stretch spanners in polylogarithmic time

The paper presents a deterministic distributed algorithm that given an n node unweighted graph constructs an O(n^3/2) edge 3-spanner for it in O(log n) time. This algorithm is then extended into a deterministic algorithm for computing an O(kn^1+1/k) edge O(k)-spanner in 2^O(k) log^k-1 n time for every integer parameter k ≥ 1. This establishes that the problem of the deterministic construction of a linear (in k) stretch spanner with few edges can be solved in the distributed setting in polylogarithmic time. The paper also investigates the distributed construction of sparse spanners with almost pure additive stretch (1 + ε, β), i.e., such that the distance in the spanner is at most 1 + ε times the original distance plus β. It is shown, for every ε > 0, that in O(ε^-1 log n) time one can deterministically construct a spanner with O(n^3/2) edges that is both a 3-spanner and a (1 + ε, 8 log n)-spanner. Furthermore, it is shown that in n^{O(1/√log n)} + O(1/ε) time one can deterministically construct a spanner with O(n ^3/2) edges which is both a 3-spanner and a (1 + ε, 4)-spanner. This algorithm can be transformed into a Las Vegas randomized algorithm with guarantees on the stretch and time, running in O(ε^-1 +log n) expected time.