Distributed Spanner Approximation

We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for distributed setting, providing both algorithmic and hardness results. main result shows that an $\alpha$-approximation directed $k$-spanner $k \geq 5$ requires $\Omega(n /\sqrt{\alpha}\log{n})$ rounds using deterministic algorithms or $\Omega(\sqrt{n }/\sqrt{\alpha}\log{n})$ randomized ones, in Congest model computing. Combined with constant-round $O(n^{\epsilon})$-approximation algorithm Local [L. Barenboim, M. Elkin, C. Gavoille, Theoret. Comput. Sci., 751 (2016), pp. 2--23], as well a polylog-round $(1+\epsilon)$-approximation we show here, our lower bounds imply strict separation between models. Notably, to best knowledge, this is first these models local approximation problem. Similarly, undirected cases implied. also prove results weighted $k$-spanners unweighted 4$ model. In addition, 2-spanner On side, apart from aforementioned $k$-spanners, contribution new construction 2-spanners uses only polynomial computations. has guaranteed ratio $O(\log(m/n))$ graph $n$ vertices $m$ edges, which matches known polynomial-time sequential [G. Kortsarz D. Peleg, J. Algorithms, 17 (1994), 222--236], tight if restrict ourselves An factor was not previously setting. The number required $O(\log{n}\log{\Delta})$ high probability, where $\Delta$ maximum degree graph. approach allows us extend work directed, weighted, client-server variants It provides dominating set problem, $O(\log{\Delta})$ ratio.