Spanners under the Hausdorff and Fréchet distances

We initiate the study of spanners under the Hausdorff and Fréchet distances. Let S be a set of points in $R^d$ and ε a non-negative real number. A subgraph H of the Euclidean graph over S is an ε-Hausdorff-spanner (resp., an ε-Fréchet-spanner) of S, if for any two points $u,v\in S$ there exists a path $P(u,v)$ in H between u and v, such that the Hausdorff distance (resp., the Fréchet distance) between $P(u,v)$ and $\overline{uv}$ is at most ε. We show that any t-spanner of a planar point-set S is a $\frac{\sqrt{t^2-1}}{2}$-Hausdorff-spanner and a $\min \{\frac{t}{2},\frac{\sqrt{5t^2-2t-3}}{4}\}$-Fréchet spanner. We also prove that for any $t>1$, there exist a set of points S and an ${\epsilon }_1$-Hausdorff-spanner of S and an ${\epsilon }_2$-Fréchet-spanner of S, where ${\epsilon }_1$ and ${\epsilon }_2$ are constants, such that neither of them is a t-spanner.