  We present a new algorithm that produces a well-spaced superset of points
conforming to a given input set in any dimension with guaranteed optimal output
size. We also provide an approximate Delaunay graph on the output points. Our
algorithm runs in expected time $O(2^{O(d)}(n\log n + m))$, where $n$ is the
input size, $m$ is the output point set size, and $d$ is the ambient dimension.
The constants only depend on the desired element quality bounds.
  To gain this new efficiency, the algorithm approximately maintains the
Voronoi diagram of the current set of points by storing a superset of the
Delaunay neighbors of each point. By retaining quality of the Voronoi diagram
and avoiding the storage of the full Voronoi diagram, a simple exponential
dependence on $d$ is obtained in the running time. Thus, if one only wants the
approximate neighbors structure of a refined Delaunay mesh conforming to a set
of input points, the algorithm will return a size $2^{O(d)}m$ graph in
$2^{O(d)}(n\log n + m)$ expected time. If $m$ is superlinear in $n$, then we
can produce a hierarchically well-spaced superset of size $2^{O(d)}n$ in
$2^{O(d)}n\log n$ expected time.
