# On Higher Order Voronoi Diagrams of Line Segments

@inproceedings{Papadopoulou2012OnHO,
title={On Higher Order Voronoi Diagrams of Line Segments},
author={Evanthia Papadopoulou and Maksym Zavershynskyi},
booktitle={ISAAC},
year={2012}
}
• Published in ISAAC 19 December 2012
• Mathematics, Computer Science
We analyze structural properties of the order-k Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single order-k Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of non-intersecting segments remains O(k(n − k)) similarly to points. For intersecting line segments…
13 Citations
Linear-Time Algorithms for the Farthest-Segment Voronoi Diagram and Related Tree Structures
• Computer Science
ISAAC
• 2015
We present linear-time algorithms to construct tree-like Voronoi diagrams with disconnected regions after the sequence of their faces along an enclosing boundary (or at infinity) is known. We focus
On the Structure of Higher Order Voronoi Cells
• Mathematics
J. Optim. Theory Appl.
• 2019
The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given $k$-element subset of the set of sites consists of the $k$ closest sites.
An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
• Computer Science, Mathematics
Algorithmica
• 2018
This study study covers many concrete order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and proposes a randomized incremental construction algorithm that runs in O(k(n-k)log2n+nlog3n) steps.
An Expected Linear-Time Algorithm for the Farthest-Segment Voronoi Diagram
• Computer Science, Mathematics
• 2014
An expected linear-time algorithm to construct the farthest-segment Voronoi diagram, given the sequence of its faces at infinity, and it can be computed in O(n log n) time, where n is the number of input segments.
On the Complexity of Higher Order Abstract Voronoi Diagrams
• Computer Science, Mathematics
ICALP
• 2013
This paper is the first to study abstract Voronoi diagrams of arbitrary order k and proves that their complexity is upper bounded by 2k(n−k), and the reasons for this bound are combinatorial properties of certain permutation sequences.
The Higher-Order Voronoi Diagram of Line Segments
• Mathematics, Computer Science
Algorithmica
• 2014
The structural properties of this Voronoi diagram are analyzed and it is shown that its combinatorial complexity is O(k(n-k), for non-crossing line segments, despite the presence of disconnected regions.
Higher-order Voronoi diagrams of polygonal objects
It is proved that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n−k), as in the case of points, and the connection between two mathematical abstractions: abstract Voronoa diagrams and the Clarkson-Shor framework is established.
A Randomized Divide and Conquer Algorithm for Higher-Order Abstract Voronoi Diagrams
• Computer Science, Mathematics
ISAAC
• 2014
A randomized divide-and-conquer algorithm to compute the order-$$k$$ abstract Voronoi diagram in expected $$O(kn^{1+\varepsilon })$$ operations is developed.
A Sweepline Algorithm for Higher Order Voronoi Diagrams
• Computer Science
2013 10th International Symposium on Voronoi Diagrams in Science and Engineering
• 2013
An algorithm to construct order-k Voronoi diagrams with a sweepline technique with O(nk2 log n) time complexity and O( nk) space complexity is presented.

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