Improved LCAs for constructing spanners

In this paper we study the problem of constructing spanners in a local manner, specifically in the Local Computation Model proposed by Rubinfeld et al. (ICS 2011). We provide an LCA for constructing $(2r-1)$-spanners with $\widetilde{O}(n^{1+1/r})$ edges and probe complexity of $\widetilde{O}(n^{1-1/r})$ $r \in \{2,3\}$, where $n$ denotes the number of vertices in the input graph. Up to polylogarithmic factors, in both cases the stretch factor is optimal (for the respective number of edges). In addition, our probe complexity for $r=2$, i.e., for constructing $3$-spanner is optimal up to polylogarithmic factors. Our result improves over the probe complexity of Parter et al. (ITCS 2019) that is $\widetilde{O}(n^{1-1/2r})$ for $r \in \{2,3\}$. For general $k\geq 1$, we provide an LCA for constructing $O(k^2)$-spanners with $\tilde{O}(n^{1+1/k})$ edges on expectation whose probe complexity is $O(n^{2/3}\Delta^2)$. This improves over the probe complexity of Parter et al. that is $O(n^{2/3}\Delta^4)$.