Light Spanners for High Dimensional Norms via Stochastic Decompositions

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

Light Spanners

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\epsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.Comment: 10 pages, 1 figure, to appear in ICALP 201

Optimal Euclidean spanners: really short, thin and lanky

In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners that achieves constant degree, diameter $O(\log n)$, and weight $O(\log^2 n) \cdot \omega(MST)$, and has running time $O(n \cdot \log n)$. This construction applies to $n$-point constant-dimensional Euclidean spaces. Moreover, Arya et al. conjectured that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time. This conjecture of Arya et al. became a central open problem in the area of Euclidean spanners. In this paper we resolve the long-standing conjecture of Arya et al. in the affirmative. Specifically, we present a construction of spanners with the same stretch, degree, diameter, and running time, as in Arya et al.'s result, but with optimal weight $O(\log n) \cdot \omega(MST)$. Moreover, our result is more general in three ways. First, we demonstrate that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics. Second, we provide a general tradeoff between the three involved parameters, which is tight in the entire range. Third, we devise a transformation that decreases the lightness of spanners in general metrics, while keeping all their other parameters in check. Our main result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4, 201