A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters

In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in the plane, the distance between a point $t$ and a cluster $P$ is measured as the maximum distance between $t$ and any point in $P$, and the diagram is defined in a nearest-neighbor sense for the input clusters. In this paper we consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for which the combinatorial complexity of the Hausdorff Voronoi diagram is linear in the total number of points, $n$, on the convex hulls of all clusters. We present a randomized incremental construction, based on point location, that computes this diagram in expected $O(n\log^2{n})$ time and expected $O(n)$ space. Our techniques efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The diagram finds direct applications in VLSI computer-aided design.Comment: arXiv admin note: substantial text overlap with arXiv:1306.583

Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region

Introducing Quantum Ricci Curvature

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centres. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behaviour for short lattices distances and compare its large-scale behaviour with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale.Comment: 43 pages, 27 figure

On the hausdorff and other cluster Voronoi diagrams

The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand various generalizations of this simple concept. While many generalized Voronoi diagrams have been well studied, many others still have unsettled questions. An example of the latter are cluster Voronoi diagrams, whose sites are sets (clusters) of objects rather than individual objects. In this dissertation we study certain cluster Voronoi diagrams from the perspective of their construction algorithms and algorithmic applications. Our main focus is the Hausdorff Voronoi diagram; we also study the farthest-segment Voronoi diagram, as well as certain special cases of the farthest-color Voronoi diagram. We establish a connection between cluster Voronoi diagrams and the stabbing circle problem for segments in the plane. Our results are as follows. (1) We investigate the randomized incremental construction of the Hausdorff Voronoi diagram. We consider separately the case of non-crossing clusters, when the combinatorial complexity of the diagram is O(n) where n is the total number of points in all clusters. For this case, we present two construction algorithms that require O(n log2 n) expected time. For the general case of arbitrary clusters, we present an algorithm that requires O((m + n log n) log n) expected time and O(m + n log n) expected space, where m is a parameter reflecting the number of crossings between clusters' convex hulls. (2) We present an O(n) time algorithm to construct the farthest-segment Voronoi diagram of n segments, after the sequence of its faces at infinity is known. This augments the well-known linear-time framework for Voronoi diagram of points in convex position, with the ability to handle disconnected Voronoi regions. (3) We establish a connection between the cluster Voronoi diagrams (the Hausdorff and the farthest-color Voronoi diagram) and the stabbing circle problem. This implies a new method to solve the latter problem. Our method results in a near-optimal O(n log2 n) time algorithm for a set of n parallel segments, and in an optimal O(n log n) time algorithm for a set of n segments satisfying some other special conditions. (4) We study the farthest-color Voronoi diagram in special cases considered by the stabbing circle problem. We prove O(n) bound for its combinatorial complexity and present an O(nlogn) time algorithm to construct it

Circles in the Water: Towards Island Group Labeling

Many algorithmic results are known for automated label placement on maps. However, algorithms to compute labels for groups of features, such as island groups, are largely missing. In this paper we address this issue by presenting new, efficient algorithms for island label placement in various settings. We consider straight-line and circular-arc labels that may or may not overlap a given set of islands. We concentrate on computing the line or circle that minimizes the maximum distance to the islands, measured by the closest distance. We experimentally test whether the generated labels are reasonable for various real-world island groups, and compare different options. The results are positive and validate our geometric formalizations

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum