Geometric $(1+\varepsilon)$-Spanners with Few Crossings

For $n$ points in the plane and an $\varepsilon>0$, we construct a $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges in which every edge has $\tilde{O}(1/\varepsilon^3)$ crossings, hence the total number of crossings is $\tilde{O}(n/\varepsilon^4)$, furthermore the ratio between the lengths of any two crossing edges is $O(1/\varepsilon^2)$. Our spanner construction substantially improves on the previous upper bound for the number of crossings in a $(1+\varepsilon)$-spanner, and it is the first spanner construction that ensures $O(1)$ crossings per edge for any constant $\varepsilon>0$. In contrast, we construct: $n$ points in the plane for which every $(1+\varepsilon)$-spanner has $\Omega(n/\varepsilon^3)$ crossings, $n$ points for which every $(1+\varepsilon)$-spanner has an edge with $\Omega(1/\varepsilon^{5/2})$ crossings, and 4 points for which every $(1+\varepsilon)$-spanner contains two crossing edges where one is $\Omega(1/\varepsilon)$ times longer than the other.