TY - JOUR

T1 - A kinetic triangulation scheme for moving points in the plane

AU - Kaplan, Haim

AU - Rubin, Natan

AU - Sharir, Micha

N1 - Funding Information:
✩ Work by Haim Kaplan and Natan Rubin was partially supported by Grant 975/06 from the Israel Science Fund. Work by Micha Sharir and Natan Rubin was partially supported by Grants 155/05 and 338/09 from the Israel Science Fund. Work by Haim Kaplan was also supported by Grant 2006-204 from the U.S.–Israel Binational Science Foundation. Work by Micha Sharir was also supported by NSF Grant CCF-08-30272, by Grant 2006-194 from the U.S.–Israel Binational Science Foundation, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Proc. 26th Annual Symposium on Computational Geometry, 2010, pp. 137–146. * Corresponding author. E-mail addresses: haimk@post.tau.ac.il (H. Kaplan), rubinnat@post.tau.ac.il (N. Rubin), michas@post.tau.ac.il (M. Sharir).

PY - 2011/5/1

Y1 - 2011/5/1

N2 - We present a simple randomized scheme for triangulating a set P of n points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of P move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme experiences an expected number of O(n2βs+2(n)log 2n) discrete changes, and handles them in a manner that satisfies all the standard requirements from a kinetic data structure: compactness, efficiency, locality and responsiveness. Here s is the maximum number of times at which any specific triple of points of P can become collinear, βs+2(q)=λ s+2 (q)/q, and λs+2(q) is the maximum length of Davenport-Schinzel sequences of order s+2 on q symbols. Thus, compared to the previous solution of Agarwal, Wang and Yu (2006) [4], we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is conceptually simpler, and easier to implement and analyze.

AB - We present a simple randomized scheme for triangulating a set P of n points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of P move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme experiences an expected number of O(n2βs+2(n)log 2n) discrete changes, and handles them in a manner that satisfies all the standard requirements from a kinetic data structure: compactness, efficiency, locality and responsiveness. Here s is the maximum number of times at which any specific triple of points of P can become collinear, βs+2(q)=λ s+2 (q)/q, and λs+2(q) is the maximum length of Davenport-Schinzel sequences of order s+2 on q symbols. Thus, compared to the previous solution of Agarwal, Wang and Yu (2006) [4], we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is conceptually simpler, and easier to implement and analyze.

KW - Kinetic data structure

KW - Pseudo-triangulation

KW - Treaps

KW - Triangulation

UR - http://www.scopus.com/inward/record.url?scp=78751650150&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2010.11.001

DO - 10.1016/j.comgeo.2010.11.001

M3 - Article

AN - SCOPUS:78751650150

SN - 0925-7721

VL - 44

SP - 191

EP - 205

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 4

ER -