TY - JOUR

T1 - Stable Delaunay Graphs

AU - Agarwal, Pankaj K.

AU - Gao, Jie

AU - Guibas, Leonidas J.

AU - Kaplan, Haim

AU - Rubin, Natan

AU - Sharir, Micha

N1 - Funding Information:
P.A. and M.S. were supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation. P.A. was also supported by NSF under Grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, and by an ERDC contract W9132V-11-C-0003. L.G. was supported by NSF grants CCF-10-11228 and CCF-11-61480. H.K. was supported by Grant 822/10 from the Israel Science Foundation, Grant 1161/2011 from the German-Israeli Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11). N.R. was supported by Grants 975/06 and 338/09 from the Israel Science Fund, by Minerva Fellowship Program of the Max Planck Society, by the Fondation Sciences Mathématiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098). M.S. was supported by NSF Grant CCF-08-30272, by Grants 338/09 and 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2015/9/3

Y1 - 2015/9/3

N2 - Let P be a set of n points in R2, and let DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of the stability of edges of DT(P). Specifically, defined in terms of a parameter α > 0, a Delaunay edge pq is called α-stable, if the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. A subgraph G of DT(P) is called a (cα, α)-stable Delaunay graph (SDG in short), for some absolute constant c ≥ 1, if every edge in G is α-stable and every cα-stable edge of DT(P) is in G. Stability can also be defined, in a similar manner, for edges of Delaunay triangulations under general convex distance functions, induced by arbitrary compact convex sets Q. We show that if an edge is stable in the Euclidean Delaunay triangulation of P, then it is also a stable edge, though for a different value of α, in the Delaunay triangulation of P under any convex distance function that is sufficiently close to the Euclidean norm, and vice-versa. In particular, a 6α-stable edge in DT(P) is α-stable in the Delaunay triangulation under the distance function induced by a regular k-gon for k ≥ 2π/α, and vice-versa. This relationship, along with the analysis in the companion paper [3], yields a linear-size kinetic data structure (KDS) for maintaining an (8α, α)-SDG as the points of P move. If the points move along algebraic trajectories of bounded degree, the KDS processes a nearly quadratic number of events during the motion, each of which can be processed in O(logn) time. We also show that several useful properties of DT(P) are retained by any SDG of P (although some other properties are not).

AB - Let P be a set of n points in R2, and let DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of the stability of edges of DT(P). Specifically, defined in terms of a parameter α > 0, a Delaunay edge pq is called α-stable, if the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. A subgraph G of DT(P) is called a (cα, α)-stable Delaunay graph (SDG in short), for some absolute constant c ≥ 1, if every edge in G is α-stable and every cα-stable edge of DT(P) is in G. Stability can also be defined, in a similar manner, for edges of Delaunay triangulations under general convex distance functions, induced by arbitrary compact convex sets Q. We show that if an edge is stable in the Euclidean Delaunay triangulation of P, then it is also a stable edge, though for a different value of α, in the Delaunay triangulation of P under any convex distance function that is sufficiently close to the Euclidean norm, and vice-versa. In particular, a 6α-stable edge in DT(P) is α-stable in the Delaunay triangulation under the distance function induced by a regular k-gon for k ≥ 2π/α, and vice-versa. This relationship, along with the analysis in the companion paper [3], yields a linear-size kinetic data structure (KDS) for maintaining an (8α, α)-SDG as the points of P move. If the points move along algebraic trajectories of bounded degree, the KDS processes a nearly quadratic number of events during the motion, each of which can be processed in O(logn) time. We also show that several useful properties of DT(P) are retained by any SDG of P (although some other properties are not).

KW - Bisector

KW - Convex distance function

KW - Delaunay triangulation

KW - Kinetic data structure

KW - Moving points

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=84945488941&partnerID=8YFLogxK

U2 - 10.1007/s00454-015-9730-x

DO - 10.1007/s00454-015-9730-x

M3 - Article

AN - SCOPUS:84945488941

SN - 0179-5376

VL - 54

SP - 905

EP - 929

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -