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Light Spanners for High Dimensional Norms via Stochastic Decompositions

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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.

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Notes

  1. That is a set of points \(X\subset \mathbb {R}^d\) equipped with the Euclidean metric \(\ell _2\), for small d.

  2. A metric space (X, d) has doubling constant \(\lambda \) if for every \(x\in X\) and radius \(r>0\), the ball B(x, 2r) can be covered by \(\lambda \) balls of radius r. The doubling dimension is defined as \(\mathrm{ddim}=\log _2\lambda \). A d-dimensional \(\ell _p\) space has \(\mathrm{ddim}=\Theta (d)\), and every n point metric has \(\mathrm{ddim}=O(\log n)\).

  3. The genus of a graph is minimal integer g, such that the graph could be drawn on a surface with g “handles”.

  4. \(d(A_i,B_i)=\max \{d(x,y)~\mid ~ x\in A_i,~y\in B_i\}\) is the maximum pairwise distance between \(A_i\) to \(B_i\).

  5. Note that there is no explicit dependence between the stretch parameter t, and the sparsity/lightness. Nonetheless, these dependence is implicit in the decomposition parameters, as for smaller stretch t, the inclusion parameter \(\delta \) becomes smaller as well, and thus the sparsity/lightness grow.

  6. Originally this fact was observed by James R. Lee and Anastasios Sidiropoulos. A proof sketch could be found in the full version of [7].

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Acknowledgements

We would like to thank an anonymous reviewer for useful comments.

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  1. Bar-Ilan University, Ramat Gan, Israel

    Arnold Filtser

  2. Ben-Gurion University of the Negev, Beersheba, Israel

    Ofer Neiman

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A preliminary version of this paper appeared in proceedings of 26th European Symposium on Algorithms (ESA 2018) [37]. This full version contains a new result on light (sub-graph) spanners for graphs in general (Theorem 4), and for graph of bounded treewidth (Corollary 7) in particular.

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Filtser, A., Neiman, O. Light Spanners for High Dimensional Norms via Stochastic Decompositions. Algorithmica 84, 2987–3007 (2022). https://doi.org/10.1007/s00453-022-00994-0

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