An Efficient Query Structure for Mesh Refinement

We are interested in the following mesh refinement problem: given an input set of points P in R, we would like to produce a good-quality triangulation by adding new points in P . Algorithms for mesh refinement are typically incremental; they compute the Delaunay triangulation of the input, then insert points one by one. However, retriangulating after each insertion can take linear time. In this work, we develop a query structure that maintains the mesh, without paying the full cost of retriangulating. Assuming that the meshing algorithm processes bad-quality simplices in increasing order of their shortest edge, our structure responds in constant time to point insertions, and can in constant time compute a restriction of the Voronoi cell of a point.