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, dH(u, v) ≤ α ·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(βn) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)-spanner of at most kn 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) edges in constant time. More interestingly, for every integer k > 1, it can construct in constant time a (1 + ε,O(1/ε))-spanner of O(εn) 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) edges, where β = k . For k = 3, it provides a (1 + ε, 6 − ε)-spanner with O(εn) 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. ⋆ Supported by the équipe-projet INRIA “DOLPHIN”. ⋆⋆ Supported by the ANR-project “ALADDIN”, and the équipe-projet INRIA “CÉPAGE”. ⋆ ⋆ ⋆ Supported by grants from the Israel Science Foundation and theMinerva Foundation. † Supported by the ANR-project “ALADDIN”, and the équipe-projet INRIA “GANG”.