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Spanners for metric spaces have been extensively studied, both in generalmetrics and in restricted classes, perhaps most notably in low-dimensionalEuclidean spaces -- due to their numerous applications. Euclidean spanners canbe viewed as means of compressing the $\binom{n}{2}$ pairwise distances of a$d$-dimensional Euclidean space into $O(n) = O_{\epsilon,d}(n)$ spanner edges,so that the spanner distances preserve the original distances to within afactor of $1+\epsilon$, for any $\epsilon > 0$. Moreover, one can compute suchspanners in optimal $O(n \log n)$ time. Once the spanner has been computed, itserves as a "proxy" overlay network, on which the computation can proceed,which gives rise to huge savings in space and other important quality measures.  On the negative side, by working on the spanner rather than the originalmetric, one loses the key property of being able to efficiently "navigate"between pairs of points. While in the original metric, one can go from anypoint to any other via a direct edge, it is unclear how to efficiently navigatein the spanner: How can we translate the existence of a "good" path into anefficient algorithm finding it? Moreover, usually by "good" path we mean a pathwhose weight approximates the original distance between its endpoints -- but apriori the number of edges (or "hops") in the path could be huge. To controlthe hop-length of paths, one can try to upper bound the spanner's hop-diameter,but naturally bounded hop-diameter spanners are more complex than spanners withunbounded hop-diameter, which might render the algorithmic task of efficientlyfinding good paths more challenging.  The original metric enables us to navigate optimally -- a single hop (for anytwo points) with the exact distance, but the price is high -- $\Theta(n^2)$edges. [...]

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners