Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional 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 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 with unbounded hop-diameter, 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. [...]

中文翻译：

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只见树木不见森林：通过有界跃点直径扳手在度量空间中导航

度量空间的 Spanner 已被广泛研究，无论是在一般度量中还是在受限类中，也许最显着的是在低维欧几里得空间中——由于它们的众多应用。欧几里得扳手可以看作是将 $\binom{n}{2}$ 维欧几里得空间的 $\binom{n}{2}$ 成对距离压缩成 $O(n) = O_{\epsilon,d}(n)$ 扳手的手段边缘，因此对于任何 $\epsilon > 0$，扳手距离将原始距离保持在 $1+\epsilon$ 的因子内。此外，我们可以在最优的 $O(n \log n)$ 时间内计算出这样的扳手。一旦计算完spanner，它就作为一个“代理”覆盖网络，可以在其上进行计算，从而节省大量空间和其他重要的质量措施。在消极方面，通过使用扳手而不是原始指标，人们失去了能够在点对之间有效“导航”的关键属性。虽然在原始度量中，可以通过直接边缘从任何点到任何其他点，但尚不清楚如何在扳手中有效导航：我们如何将“好”路径的存在转化为找到它的有效算法？此外，通常所说的“好”路径是指权重接近其端点之间原始距离的路径——但先验地，路径中的边（或“跳数”）数量可能很大。为了控制路径的跳跃长度，可以尝试设置扳手的跳跃直径上限，但是自然有界跳跃直径的扳手比具有无限跳跃直径的扳手更复杂，这可能会使有效地找到好的路径的算法任务更具挑战性。原始指标使我们能够以最佳方式导航——具有精确距离的单跳（对于任意两点），但代价高昂——$\Theta(n^2)$ 边。[...]