Approximating the minimum weight steiner triangulation

Abstract

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher dimensional triangulation problems. No previous polynomial time triangulation algorithm was known to approximate the MWST within a factor better than O(log n).

Parallel construction of quadtrees and quality triangulations

“…3 Moreover, the total edge length can be made to approximate that of the minimum-weight triangulation. 4 The following theorem extends this result to the parallel case.…”

“…Eppstein 4 showed how to sequentially compute triangulations of point sets with these guarantees: all angles between 36 • and 80 • , total edge length within a constant factor of the minimum, and total number of triangles within a constant of the minimum for any angle-bounded triangulation. The algorithm again uses local warping, trivial to parallelize, but the quadtree must also satisfy some stronger conditions than the ones given directly by Theorems 1 and 2:…”

Squarepants in a tree

“…We note that a similar argument based on local feature sizes can be used to provide an alternative proof for our previous result [14] that quadtree triangulations have total length within a constant factor of the minimum length Steiner triangulation of a point set.…”

Minimum Weight Convex Steiner Partitions

“…Clarkson [10] proved that any set of n points in the plane admits a Steiner triangulation of weight O(W log n), and Eppstein [14] showed that this bound is the best possible. His construction consists of 4 vertices of a square and n − 4 points evenly distributed along a circle placed in the interior of the square, see Figure 1(b).…”