Can't See the Forest for the Trees

Omri Kahalon, Hung Le, Lazar Milenković, Shay Solomon
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
Spanners for metric spaces have been extensively studied, perhaps most notably in low-dimensional Euclidean spaces -due to their numerous applications. Euclidean spanners can be viewed as means of compressing the 𝑛 2 pairwise distances of a 𝑑-dimensional Euclidean space into 𝑂 (𝑛) = 𝑂 𝜖,𝑑 (𝑛) spanner edges, so that the spanner distances preserve the original distances to within a factor of 1 + 𝜖, for any 𝜖 > 0. Moreover, one can compute such spanners efficiently in the standard centralized and
more » ... istributed settings. 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. The original metric enables us to "navigate" optimally -a single hop (for any two points) with the exact distance, but the price is high -Θ(𝑛 2 ) edges. Is it possible to efficiently navigate, on a sparse spanner, using 𝑘 hops and approximate distances, for 𝑘 close to 1 (say 𝑘 = 2)? Surprisingly, this fundamental question has been overlooked in Euclidean spaces, as well as in other classes of metrics, despite the long line of work on spanners in metric spaces. We answer this question in the affirmative via a surprisingly simple observation on bounded hop-diameter spanners for tree metrics, which we apply on top of known, as well as new, tree cover theorems. Beyond its simplicity, the strength of our approach is three-fold: • Applicable: We present a variety of applications of our efficient navigation scheme, including a 2-hop routing scheme in Euclidean spaces with stretch 1 + 𝜖 using 𝑂 (log 2 𝑛) bits of memory for labels and routing tables -to the best of our knowledge, all known routing schemes prior to this work use Ω(log 𝑛) hops. • Unified: Our navigation scheme and applications extend beyond Euclidean spaces to any class of metrics that admits an
doi:10.1145/3519270.3538414 fatcat:oynmomltr5aqbdc56pi7fvfgla