Close the open intermediate-distance gap for sparse spanners

Determine the optimal stretch guarantees for sparse $(\alpha,\beta)$-spanners in the intermediate distance range around $d\approx k$, thereby resolving the remaining gap between the known bounds of $f(d)=O_\varepsilon(k+d)$ outside the window $d\in[k^{1-\varepsilon},k^{1+\varepsilon}]$ and the target behavior near $d=k$.

Background

The paper surveys prior constructions of near-optimal sparse $(\alpha,\beta)$-spanners and notes that the known guarantees do not fully characterize the intermediate distance regime near $d=k$. Ben-Levy and Parter obtain nearly optimal bounds outside a window centered at $k$, with specific guarantees for shorter distances and constant multiplicative stretch only for sufficiently large distances.

The paper’s construction improves the short-distance guarantee to $2k+O(d\log d)$ and obtains an $(O(\log k),O(k))$-spanner, but it does not achieve the conjectured constant multiplicative stretch with an $O(k)$ additive term around $d\approx k$. Thus, the unresolved issue is the remaining gap in the stretch-versus-sparsity tradeoff in that intermediate range.