77 research outputs found
An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons
Given a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P. It is known that the problem has an Ω(n + m log m) time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that can solve the problem in O(n + m log m) expected time. The previous best deterministic algorithms solve the problem in O(n log log n + m log m) time [Oh, Barba, and Ahn, SoCG 2016] or in O(n + m log m + m log2 n) time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm of O(n + m log m) time, which is optimal. This answers an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before
Visibility-based coverage of mobile sensors in non-convex domains
The area coverage problem of mobile sensor networks has attracted much attention recently, as mobile sensors find many important applications in remote and hostile environments. However, the deployment of mobile sensors in a non-convex domain is nontrivial due to the more general shape of the domain and the attenuation of sensing capabilities caused by the boundary walls or obstacles. We consider the problem of exploration and coverage by mobile sensors in an unknown non-convex domain. We propose the definition of 'visibility-based Voronoi diagram' and extend the continuous-time Lloyd's method, which only works for convex domains, to deploy the mobile sensors in the unknown environments in a distributed manner. Our simulations show the effectiveness of the proposed algorithms. © 2011 IEEE.published_or_final_versionThe 8th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD2011), Qingdao, China, 28-30 June 2011. In Proceedings of the 8th ISVD, 2011, p. 105-11
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