Balancing Degree, Diameter, and Weight in Euclidean Spanners

Abstract

In a seminal paper from 1995, Arya et al. [Euclidean spanners: Short, thin, and lanky, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM, New York, 1995, pp. 489--498] devised a construction that, for any set $S$ of $n$ points in $\mathbb R^d$ and any $\epsilon > 0$, provides a $(1+\epsilon)$-spanner with diameter $O(\log n)$, weight $O(\log^2 n) \cdot w(MST(S))$, and constant maximum degree. Another construction from the same work provides a $(1+\epsilon)$-spanner with $O(n)$ edges and diameter $O(\alpha(n))$, where $\alpha$ stands for the inverse Ackermann function. There are also a few other known constructions of $(1+\epsilon)$-spanners. Das and Narasimhan [A fast algorithm for constructing sparse Euclidean spanners, in Proceedings of the 10th Annual ACM Symposium on Computational Geometry (SOCG), ACM, New York, 1994, pp. 132--139] devised a construction with constant maximum degree and weight $O(w(MST(S)))$, but the diameter may be arbitrarily large. In another construction by Arya et al., there is diameter $O(\log n)$ and weight $O(\log n) \cdot w(MST(S))$, but this construction may have arbitrarily large maximum degree. While these constructions address some important practical scenarios, they fail to address situations in which we are prepared to compromise on one of the parameters but cannot afford for this parameter to be arbitrarily large. In this paper we devise a novel unified construction that trades gracefully among the maximum degree, diameter, and weight. For a positive integer $k$ our construction provides a $(1+\epsilon)$-spanner with maximum degree $O(k)$, diameter $O(\log_k n + \alpha(k))$, weight $O(k \cdot \log_k n \cdot \log n) \cdot w(MST(S))$, and $O(n)$ edges. Note that for $k = O(1)$ this gives rise to maximum degree $O(1)$, diameter $O(\log n)$, and weight $O(\log^2 n) \cdot w(MST(S))$, which is one of the aforementioned results of Arya et al. For $k= n^{1/\alpha(n)}$ this gives rise to diameter $O(\alpha(n))$, weight $O(n^{1/\alpha(n)} \cdot \log n \cdot \alpha(n)) \cdot w(MST(S))$, and maximum degree $O(n^{1/\alpha(n)})$. In the corresponding result from Arya et al., the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our bound of $O(\log_k n + \alpha(k))$ on the diameter is optimal under the constraints that the maximum degree is $O(k)$ and the number of edges is $O(n)$. Similarly to the bound of Arya et al., our bound on the weight is optimal up to a factor of $\log n$. Our construction also provides a similar trade-off in the complementary range of parameters, i.e., when the weight should be smaller than $\log^2 n$, but the diameter is allowed to grow beyond $\log n$. Moreover, all our results apply to doubling metrics. En route to these results we devise optimal constructions of 1-spanners for general tree metrics, and we employ them to build our Euclidean spanners. Subsequent papers have utilized our constructions of 1-spanners for tree metrics to resolve a long-standing conjecture of Arya et al.