### Local Computation of Nearly Additive Spanners [chapter]

Bilel Derbel, Cyril Gavoille, David Peleg, Laurent Viennot
2009 Lecture Notes in Computer Science
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
more » ... ≥ 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 kn 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(log k/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.