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We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality.
Our comparison based algorithm runs in time $O(n\log n + m)$, where $n$ is the input size and $m$ is the output size, and with constants depending only on the dimension and the desired element quality bounds.
It can terminate early in $O(n\log n)$ time returning a $O(n)$ size Voronoi diagram of a superset of $P$ with a relaxed quality bound, which again matches the known lower bounds.
The previous best results in the comparison model depended on the log of the <b>spread</b> of the input, the ratio of the largest to smallest pairwise distance among input points.
We reduce this dependence to $O(\log n)$ by using a sequence of $\epsilon$-nets to determine input insertion order in an incremental Voronoi diagram.
We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.