Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Given a pair of parameters α ≥ 1,β ≥ 0, a subgraph G'=(V,H) of an n-vertex unweighted undirected graph G=(V,E) is called an (α,β)-spanner if for every pair u,ν ∈ V of vertices, we have dG' (u,ν)≤ α dG (u,α)+β. If β=0 the spanner is called a multiplicative α-spanner, and if α = 1+ε, for an arbitrarily small ε>0, the spanner is said to be near-additive. Graph spanners [5,36], are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Near-additive spanners, introduced in [27], preserve large distances much more faithfully than the more traditional multiplicative spanners. Also, recent lower bounds [1] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with α=1), and therefore showed that essentially near-additive spanners provide the best approximation of distances that one can hope for. Numerous distributed algorithms, for constructing sparse near-additive spanners, were devised in [17,20,25,28,40]. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [25]., and also there are efficient deterministic algorithms in the LOCAL model [17]. However, the only known deterministic CONGEST-model algorithm for the problem [20] requires super-linear time in n. In this paper, we remedy the situation and devise an efficient deterministic CONGEST-model algorithm for constructing arbitrarily sparse near-additive spanners. The running time of our algorithm is low polynomial, i.e., roughly O(β ⋅ nρ), where ρ > 0 is an arbitrarily small positive constant that affects the additive term β. In general, the parameters of our new algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm due to [25].