Deformable spanners and applications

Abstract

For a set S of points in R d , an s-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d ) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers.

Kinetic Facility Location

“…The centers that they considered are a subset of the moving nodes, whereas Bespamyatnikh et al [14] studied k-center problems for k = 1 in the KDS framework, where the centers are not necessarily located at the moving points. Another algorithm for the kinetic k-center problem can be found in [19]. Hershberger [26] proposed a kinetic algorithm for maintaining a covering of the moving points in R d by unit boxes such that the number of boxes is always within a factor of 3 d of the optimal static covering at any instance.…”

Fully Dynamic Geometric Spanners

“…In [6], Bose, Gudmundsson and Morin presented an algorithm that supports only insertions in a polylogarithmic time. Most recently Gao, Guibas and Nguyen [10] considered dynamic and kinetic geometric spanners. They obtained a fully dynamic geometric spanner with a worst case update time of O(log α), where α is the aspect ratio of the set S. The aspect ratio of a set of points S is the ratio of the maximum pairwise distance to the minimum pairwise distance of points in S. The only demand that the metric space has to fulfill in their paper is that it will have a constant doubling dimension.…”

“…As in Gao et al. [10] the only requirement that the metric space has to fulfill is to have a constant doubling dimension.…”

“…The spanner presented by Gao et al in [10] is based on an efficient maintenance of a hierarchical structure of the points in S. Independently, a similar hierarchical structure was used by Krauthgamer and Lee in [13] for proximity search. The time bounds of their dynamic updates depend also in the aspect ratio.…”

“…In any stage our spanner can be a spanner that was constructed by the algorithm of Gao et al in [10]. However, the insertion of a new point in our algorithm is done in a more efficient way.…”