A kinetic triangulation scheme for moving points in the plane

13 years 9 months ago

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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(n2 βs+2(n) log2 n) 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 where any speciﬁc triple of points of P can become collinear, βs+2(q) = λs+2(q)/q, and λs+2(q) is the maximum length of Davenport-Schinzel sequences of order s + 2 on n symbols. Thus, compared to the previous solution of Agarwal et al. [4], we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is simpler to implement and analyze. Categorie...

Haim Kaplan, Natan Rubin, Micha Sharir

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