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 \(\tilde{O}(n^{1+1/t^2})\) 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 \(\ell _p\) with \(1<p\le 2\). 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 \(\ell _p\) for \(1<p\le 2\) has an O(t)-spanner with \(n^{1+\tilde{O}(1/t^p)}\) edges and lightness \(n^{\tilde{O}(1/t^p)}\). 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 \(\nu =\nu (t)\) admit an O(t)-spanner with lightness \(\tilde{O}(\nu ^{1/t})\). For example, metrics with doubling constant \(\lambda \), graphs of genus g, and graphs of treewidth k, all have spanners with stretch O(t) and lightness \(\tilde{O}(\lambda ^{1/t})\), \(\tilde{O}(g^{1/t})\), \(\tilde{O}(k^{1/t})\) respectively. While these families do admit a (\(1+\epsilon \))-spanner, its lightness depend exponentially on the dimension (resp. \(\log g\), k). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.