Kinetic Geodesic Voronoi Diagrams in a Simple Polygon

02/14/2020
by   Matias Korman, et al.
0

We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m^3), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log^2 m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 6

page 17

page 21

09/24/2021

Dynamic Data Structures for k-Nearest Neighbor Queries

Our aim is to develop dynamic data structures that support k-nearest nei...
03/07/2022

Farthest-point Voronoi diagrams in the presence of rectangular obstacles

We present an algorithm to compute the geodesic L_1 farthest-point Voron...
03/09/2018

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram in a Simple Polygon

The geodesic Voronoi diagram of m point sites inside a simple polygon of...
05/21/2019

Voronoi diagram of orthogonal polyhedra in two and three dimensions

Voronoi diagrams are a fundamental geometric data structure for obtainin...
02/26/2022

Utility and Feasibility of a Center Surround Event Camera

Standard dynamic vision sensor (DVS) event cameras output a stream of sp...
12/06/2021

Voronoi Diagrams in the Hilbert Metric

The Hilbert metric is a distance function defined for points lying withi...
10/20/2019

Worst-Case Polylog Incremental SPQR-trees: Embeddings, Planarity, and Triconnectivity

We show that every labelled planar graph G can be assigned a canonical e...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.