On algorithmic complexity of imprecise spanners

Let $t>1$ be a real number. A geometric t-spanner is a geometric graph for a point set in $R^d$ with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t.

An imprecise point set is modeled by a set R of regions in $R^d$. If one chooses a point inside each region of R, then the resulting point set is called a precise instance from R. An imprecise t-spanner for an imprecise point set R is a graph $G=(R,E)$ such that for each precise instance S from R, graph $G_S=(S,E_S)$, where $E_S$ is the set of edges corresponding to E and S, is a t-spanner.

In this paper, we show an imprecise point set R of n straight-line segments in the plane such that any imprecise t-spanner for R has $\Omega \left(n^2\right)$ edges. Then, we give an algorithm that computes an imprecise t-spanner for a set of n pairwise disjoint d-dimensional balls with arbitrary sizes. This imprecise t-spanner has $O(n/{(t-1)}^d)$ edges and can be computed in $O(n\log n/{(t-1)}^d)$ time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.