Light Spanners for High Dimensional Norms via Stochastic Decompositions

Abstract

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with Õ(n 1+1/t 2 ) edges, little is known.In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of p with 1 < p ≤ 2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of p for 1 < p ≤ 2 has an O(t)-spanner with n 1+ Õ(1/t p ) edges and lightness n Õ(1/t p ) .In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)spanner with lightness O(n 1/t ). We exhibit the following tradeoff: metrics with decomposability parameter ν = ν(t) admit an O(t)-spanner with lightness Õ(ν 1/t ). For example, n-point Euclidean metrics have ν ≤ n 1/t , metrics with doubling constant λ have ν ≤ λ, and graphs of genus g have ν ≤ g. While these families do admit a (1 + )-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Covering metric spaces by few trees

“…We note that the trees obtained here are in fact ultrametrics 1 . This result provides a special type of spanner with distortion O(α) and O(n • λ 1/α • log λ • log α) edges, which improves the recent spanner construction of [28], whose number of edges was larger by a factor of O(log λ n) (though their spanner has additionally bounded lightness). The lower bound mentioned above for doubling spaces implies that our tree cover bounds cannot be substantially improved.…”

Hop-Constrained Metric Embeddings and their Applications

“…1 In this paper we require the partition property of SPCS for metric embeddings into ℓ∞, and for the oblivious buy-at-bulk. This property is also crucial in the construction of ultrametric covers [FL22a,Fil23,FGN23].…”