Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded HopDiameter Spanners
Abstract
Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in lowdimensional Euclidean spaces  due to their numerous applications. Euclidean spanners can be 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 a factor of $1+\epsilon$, for any $\epsilon > 0$. Moreover, one can compute such spanners in optimal $O(n \log n)$ time. Once the spanner has been computed, it serves 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 original metric, one loses the key property of being able to efficiently "navigate" between pairs of points. While in the original metric, one can go from any point to any other via a direct edge, it is unclear how to efficiently navigate in the spanner: How can we translate the existence of a "good" path into an efficient algorithm finding it? Moreover, usually by "good" path we mean a path whose weight approximates the original distance between its endpoints  but a priori the number of edges (or "hops") in the path could be huge. To control the hoplength of paths, one can try to upper bound the spanner's hopdiameter, but naturally bounded hopdiameter spanners are more complex than spanners with unbounded hopdiameter, which might render the algorithmic task of efficiently finding good paths more challenging. The original metric enables us to navigate optimally  a single hop (for any two points) with the exact distance, but the price is high  $\Theta(n^2)$ edges. [...]
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.14221
 arXiv:
 arXiv:2107.14221
 Bibcode:
 2021arXiv210714221K
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Geometry
 EPrint:
 Abstract truncated to fit arXiv limits