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...

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Bibliographic Details
Published in:Algorithmica Vol. 81; no. 6; pp. 2317 - 2345
Main Authors: Bohler, Cecilia, Klein, Rolf, Liu, Chih-Hung
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2019
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Axioms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Mathematics of Computing
Randomization
Theory of Computation
Voronoi graphs
Order
Computational geometry
Voronoi diagrams
Randomized geometric algorithms
Abstract Voronoi diagrams
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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. 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, a running time with a polylog factor to the number of faces was only achieved for point sites in the L 1 or Euclidean metric before.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-018-00536-7