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A Kinetic Triangulation Scheme for Moving Points in The Plane ∗
"... 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 ..."
Abstract

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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(n 2 βs+2(n) log 2 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 DavenportSchinzel 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. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Geometrical problems and computations; G.2.1 [Discrete mathematics]: Combinatorics—Combinatorial algorithms
Kinetic Stable Delaunay Graphs
"... The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in R 2 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 ..."
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The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in R 2 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 ∗ (n 2) 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. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Geometrical problems and computations; G.2.1 [Discrete mathematics]: Combinatorics— Combinatorial algorithms