TY - GEN

T1 - Kinetic stable Delaunay graphs

AU - Agarwal, Pankaj K.

AU - Gao, Jie

AU - Guibas, Leonidas J.

AU - Kaplan, Haim

AU - Koltun, Vladlen

AU - Rubin, Natan

AU - Sharir, Micha

PY - 2010/7/30

Y1 - 2010/7/30

N2 - The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℝ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter α > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O* (n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/α and polylogarithmic in n. The first structure is simpler but the dependency on 1/α in its performance is higher.

AB - The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℝ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter α > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O* (n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/α and polylogarithmic in n. The first structure is simpler but the dependency on 1/α in its performance is higher.

KW - Delaunay triangulation

KW - Kinetic data structures

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=77954924604&partnerID=8YFLogxK

U2 - 10.1145/1810959.1810984

DO - 10.1145/1810959.1810984

M3 - Conference contribution

AN - SCOPUS:77954924604

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 127

EP - 136

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

T2 - 26th Annual Symposium on Computational Geometry, SoCG 2010

Y2 - 13 June 2010 through 16 June 2010

ER -