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Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

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Pankaj K. Agarwal, Haim Kaplan, Natan Rubin, Micha Sharir

Computational Geometry · Submitted 18 Apr 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points.

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arXiv ID1404.4851 (version 1)
Pith ID2XHATFVZ, Pith's permanent record of this paper and its review
CitationHaim Kaplan, Micha Sharir, Natan Rubin, Pankaj K. Agarwal. "Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions." arXiv:1404.4851 (2014). https://pith.science/paper/2XHATFVZ
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,853 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. background 1.00 0.11 0.02 asserted 0.53
s1 We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. contribution 0.99 0.92 0.11 asserted 0.98
s2 Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. result 0.99 0.95 0.78 asserted 1.00
s3 Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework. contribution 0.65 0.94 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

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