Distributed Spanner Approximation

We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result shows that an α-approximation for the minimum directed kspanner problem for k ≥ 5 requires Ω(n/ √ α log n) rounds using deterministic algorithms or Ω( √ n/ √ α log n) rounds using randomized ones, in the Congest model of distributed computing. Combined with the constant-round O(n )-approximation algorithm in the Local model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round (1+ )-approximation algorithm in the Local model that we show here, our lower bounds for the Congest model imply a strict separation between the Local and Congest models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove that the minimum weighted k-spanner problem for k ≥ 4 requires a near-linear number of rounds in the Congest model, for directed or undirected graphs. In addition, we show lower bounds for the minimum weighted 2-spanner problem in the Congest and Local models. On the algorithmic side, apart from the aforementioned (1 + )-approximation algorithm for minimum k-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of O(log(m/n)) for a graph with n vertices and m edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. An algorithm with this approximation factor was not previously known for the distributed setting. The number of rounds required for our algorithm is O(log n log ∆) w.h.p, where ∆ is the maximum degree in the graph. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem. It also provides a Congest algorithm for the minimum dominating set problem, with a guaranteed O(log ∆) approximation ratio. ∗Technion, Department of Computer Science, {ckeren,smichald}@cs.technion.ac.il. Supported in part by the Israel Science Foundation (grant 1696/14). ar X iv :1 80 2. 03 16 0v 1 [ cs .D S] 9 F eb 2 01 8