Untangling triangulations through local explorations
Proceedings of the twenty-fourth annual symposium on Computational geometry
Kinetic stable Delaunay graphs
Proceedings of the twenty-sixth annual symposium on Computational geometry
A kinetic triangulation scheme for moving points in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
A kinetic triangulation scheme for moving points in the plane
Computational Geometry: Theory and Applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
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A triangulation of a set S of points in the plane is a subdivision of the convex hull of S into triangles whose vertices are points of S. Given a set S of n points in ${\Bbb R}^2,$ each moving independently, we wish to maintain a triangulation of S. The triangulation needs to be updated periodically as the points in S move, so the goal is to maintain a triangulation with a small number of topological events, each being the insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes $n^2 2^{O(\sqrt{\log n \cdot \log \log n})}$ topological events with high probability if the trajectories of input points are algebraic curves of fixed degree. Each topological event can be processed in $O(\log n)$ time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events, and almost matches the $\Omega(n^2)$ lower bound [1]. The number of topological events can be reduced to $nk\cdot 2^{O(\sqrt{\log k \cdot \log\log n})}$ if only k of the points are moving.