Local Computation of Nearly Additive Spanners

Abstract

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, dH(u, v) ≤ α ċ dG(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 k ≥ 1, an (α, β)-spanner of O(βn1+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 kn1+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 Ɛ > 0, it can also produce a (1 + Ɛ, 2 - Ɛ)-spanner of O(n3/2) edges in constant time. More interestingly, for every integer k > 1, it can construct in constant time a (1 + Ɛ, O(1/Ɛ)k-2)-spanner of O(Ɛ-k+1n1+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 + Ɛ, β)-spanner with O(βn1+1/k) edges, where β = klog(log k/Ɛ)+O(1). For k = 3, it provides a (1 + Ɛ, 6 - Ɛ)-spanner with O(Ɛ-1n4/3) edges. The additive terms β = β(k, Ɛ) in the stretch of our constructions yield the best trade-off currently known between k and Ɛ, 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.