No abstract

Let P be a collection of n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants s, c > 0 such that any four points are co-circular at most s times, and any three points are collinear at most c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during the motion of the points of P. In this paper, we obtain an upper bound of O(n 2+ε), for any ε > 0, under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.

We consider the problem of maintaining the Euclidean Delaunay triangulation DT of a set P of n moving points in the plane, along algebraic tranjectories of constant description complexity. Since the best known upper bound on the number of topological changes in the full Delaunay triangulation is only nearly cubic, we seek to maintain a suitable portion of the diagram that is less volatile yet retains many useful properties of the full triangulation. We introduce the notion of a stable Delaunay graph, which is a dynamic subgraph of the Delaunay triangulation. The stable Delaunay graph (a) is easy to define, (b) experiences only a nearly quadratic number of discrete changes, (c) is robust under small changes of the norm, and (d) possesses certain useful properties for further applications.The stable Delaunay graph (SDG in short) 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 e pq are at least α. We show that (i) SDG always contains at least roughly one third of the Delaunay edges at any fixed time; (ii) it contains the β-skeleton of P , for β = 1 + Ω(α 2 ); (iii) it is stable, in the sense that its edges survive for long periods of time, as long as the orientations of the segments connecting (nearby) points of P do not change by much; and (iv) stable Delaunay edges remain stable (with an appropriate redefinition of stability) if we replace the Euclidean norm by any sufficiently close norm.In particular, if we approximate the Euclidean norm by a polygonal norm (with a regular k-gon as its unit ball, with k = Θ(1/α)), we can define and keep track of a Euclidean SDG by maintaining the full Delaunay triangulation of P under the polygonal norm (which is trivial to do, and which is known to involve only a nearly quadratic number of discrete changes).We describe two kinetic data structures for maintaining SDG when the points of P move along pseudo-algebraic trajectories of constant description complexity. The first uses the polygonal norm approximation noted above, and the second is slightly more involved, but significantly reduces the dependence of its performance on α. Both structures use O * (n) storage and process O * (n 2 ) events during the motion, each in O * (1) time. (Here the O * (·) notation hides multiplicative factors which are polynomial in 1/α and polylogarithmic in n.) * A preliminary version of this paper appeared in

We show that for any finite point set P in the plane and > 0 there exist O 1 3/2+γ points in R 2 , for arbitrary small γ > 0, that pierce every convex set K with |K ∩ P | ≥ |P |. This is the first improvement of the bound of O 1 2 that was obtained in 1992 by Alon, Bárány, Füredi and Kleitman for general point sets 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 experiences an expected number of O(n 2 β s+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, β s+2 (q) = λ s+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.