Dynamic well-spaced point sets

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 O(logΔ) time, where Δ is the geometric spread, a natural measure that yields an O(logn) bound when input points are represented by log-size words. We show that this algorithm is time-optimal by proving a lower bound of Ω(logΔ) 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.