Local computation of nearly additive spanners

An (α,β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u,v, d _H (u,v) ≤α•d _G (u,v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α,β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every n-node graph and integer kge;1, an (α,β)- spanner of O(βn ^1+1/k ) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k-1,0)-spanner of at most k n ^1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC '08). For k=2 and constant ε>0, it can also produce a (1+ε,2-ε)-spanner of O(n ^3/2) edges in constant time. More interestingly, for every integer k>1, it can construct in constant time a (1+ε,O(1/ε) ^k-2)-spanner of O(ε ^-k+1 n ^1+1/k ) edges. Such deterministic construction was not previously known. The paper also presents a second generic deterministic and distributed algorithm based on the construction of small dominating sets and maximal independent sets. After computing such sets in sub-polynomial time, it constructs at its best a (1+ε,β)-spanner with O(βn ^1+1/k ) edges, where β=k ^{log(logk/ε)+O(1)}. For k=3, it provides a (1+ε, 6-ε)-spanner with O(ε ^-1 n ^4/3) edges. The additive terms β=β(k,ε) in the stretch of our constructions yield the best trade-off currently known between k and ε, due to Elkin and Peleg (STOC '01). Our distributed algorithms are rather short, and can be viewed as a unification and simplification of previous constructions.