Date of Original Version
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at http://dx.doi.org/10.1016/j.comgeo.2012.11.007
Abstract or Description
In a well-spaced point set the Voronoi cells all have bounded aspect ratio. Well-spaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that are important in scientific computations. In this paper, we consider the dynamic well-spaced point set problem, which requires constructing a well-spaced superset of a dynamically changing input set, e.g., as input points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in time, where Δ is the geometric spread, a natural measure that yields an bound when input points are represented by log-size words. We show that this algorithm is time-optimal by proving a lower bound of for a dynamic update. We also show that this algorithm maintains size-optimal outputs: the well-spaced supersets are within a constant factor of the minimum possible size. The asymptotic bounds in our results work in any constant dimensional space. Experiments with a preliminary implementation indicate that dynamic changes may be performed with considerably greater efficiency than re-constructing a well-spaced point set from scratch. To the best of our knowledge, these are the first time- and size-optimal algorithms for dynamically maintaining well-spaced point sets.
Computational Geometry, 46, 6, 756-773.