Light Euclidean Steiner Spanners in the Plane

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in $\mathbb{R}^d$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\varepsilon>0$ and $d\in \mathbb{N}$ of the minimum lightness of $(1+\varepsilon)$-spanners, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner $(1+\varepsilon)$-spanners of lightness $O(\varepsilon^{-1}\log\Delta)$ in the plane, where $\Delta\geq \Omega(\sqrt{n})$ is the \emph{spread} of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness $\tilde{O}(\varepsilon^{-(d+1)/2})$ in dimensions $d\geq 3$. Recently, Bhore and T\'{o}th (2020) established a lower bound of $\Omega(\varepsilon^{-d/2})$ for the lightness of Steiner $(1+\varepsilon)$-spanners in $\mathbb{R}^d$, for $d\ge 2$. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions $d\geq 2$. In this work, we show that for every finite set of points in the plane and every $\varepsilon>0$, there exists a Euclidean Steiner $(1+\varepsilon)$-spanner of lightness $O(\varepsilon^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.Comment: 29 pages, 14 figures. A 17-page extended abstract will appear in the Proceedings of the 37th International Symposium on Computational Geometr

Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.Comment: conference version in Symposium on Theory of Computing (1994

Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

We evaluate the virial coefficients B_k for k<=10 for hard spheres in dimensions D=2,...,8. Virial coefficients with k even are found to be negative when D>=5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D>=5. Further analysis provides evidence that negative virial coefficients will be seen for some k>10 for D=4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D=3.Comment: 33 pages, 12 figure

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Fully dynamic maintenance of euclidean minimum spanning trees

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path