38 research outputs found
On Euclidean Steiner (1+?)-Spanners
Lightness and sparsity are two natural parameters for Euclidean (1+?)-spanners. Classical results show that, when the dimension d ? ? and ? > 0 are constant, every set S of n points in d-space admits an (1+?)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ? > 0 for constant d ? ? have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+?)-spanner. They gave upper bounds of O?(?^{-(d+1)/2}) for the minimum lightness in dimensions d ? 3, and O?(?^{-(d-1))/2}) for the minimum sparsity in d-space for all d ? 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+?)-spanners of lightness O(?^{-1}log?) in the plane, where ? ? ?(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points.
In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+?)-spanners. Using a new geometric analysis, we establish lower bounds of ?(?^{-d/2}) for the lightness and ?(?^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ? 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+?)-spanners of lightness O(?^{-1}log n) for n points in Euclidean plane
Space-Efficient Algorithms for Reachability in Directed Geometric Graphs
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem.
For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m^{1/2} log n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m^{1/2} log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ? > 0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n^{1/4+?}) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques
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 . In a recent breakthrough, Le and Solomon
(2019) established the precise dependencies on and of the minimum lightness of -spanners, and
observed that additional Steiner points can substantially improve the
lightness. Le and Solomon (2020) constructed Steiner -spanners
of lightness in the plane, where 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 in
dimensions . Recently, Bhore and T\'{o}th (2020) established a lower
bound of for the lightness of Steiner
-spanners in , for . The central open
problem in this area is to close the gap between the lower and upper bounds in
all dimensions .
In this work, we show that for every finite set of points in the plane and
every , there exists a Euclidean Steiner
-spanner of lightness ; this matches the
lower bound for . 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