1. Localizing the Delaunay Triangulation and its Parallel Implementation
Good morning everyone, I’m Renjie Chen from Technion, the Israel institute of technology.
In the following, I’ll talk about my latest joint work with Craig Gotsman
about Localizing the Delaunay Triangulation
and its Parallel Implementation
Renjie Chen Craig Gotsman
Technion – Israel Institute of Technology

2. Outline Introduction of Delaunay triangulation(DT) Localize DT
Parallel DT
Experimental results & conclusion
2018/11/17
Here’s the outline of this talk.
First, I will give a very brief introduction for the Delaunay triangulation.
Next, I will talk about a local property about Delaunay triangulation.
And based on this local property, I will present a parallel algorithm for computing the Delaunay triangulation
Finally, I will show some experimental results and conclude the talk.

3. Delaunay Triangulation
2018/11/17
Given a point set, the Delaunay triangulation is a partition of the domain with triangles, which do not overlap with each other.
And each triangle has a circumcircle which doesn’t contain any other point

4. Delaunay Triangulation - Voronoi Diagram
2018/11/17
The Delaunay triangulation has a dual, the Voronoi Diagram
Each vertex of the Voronoi diagram is the circumcenter of some Delaunay triangle.
By connecting the circumcircles properly, we have the Voronoi diagram

5. Delaunay Triangulation - Algorithms
2018/11/17
Edge flip non-Delaunay triangulation
(Randomized) Incremental construction
Divide and conquer
Sweepline
Sweephull
Half-plane intersecting …
There are many algorithms developed to compute the Delaunay triangulation
To name a few,
We have edge flipping algorithm based on a non-Delaunay triangulation
Incremental construction algorithm and its randomized variant
Divide and conquer
Sweepline and sweephull
And lastly half-plane intersecting, based on which we develop the parallel Delaunay triangulation algorithm

6. Delaunay Triangulation - Properties
Empty circumcircle
Convex hull
Nearest neighbor …
2018/11/17
Delaunay triangulation has many properties, among them we have
The empty circumcircle property we mentioned earlier
Convex hull property which says the Delaunay triangulaiton always contains the convex hull of the same point set
And the nearest neighbor graph property, which says the point nearest to each point is a neighbor in the Delaunay triangulation
Our local Delaunay property is closely related to the last property

7. Delaunay Triangulation
2018/11/17
There is a lower bound on the distance of Delaunay neighbors, which is the minimal distance from a point to all the other points
However we can not bound the distance from up. That is the maximal Delaunay edge length can be as large as the diameter of the point set.
This happen most often near the convex hull.
Now the problem is can we localize all the Delaunay neighbors in some way?
Delaunay neighbor can be far away
Can we localize it?

8. Localize Delaunay Triangulation
2018/11/17
For a given point, shown red
If we are given a containing polygon formed by some points in the point set
Then we can check the triangles formed by each edge of the polygon and the given point
each such triangle has a circumcircle, shown transparently here
And the union of these circumcircles defines the region which contains all the Delaunay neighbors of the current point.
For example, the sub set shown in blue contains all the Delaunay neighbors of the red point,
And none of the black points can be Delaunay neighbors of the red point.
To verify, we can see the true Delaunay one-ring, colored in light blue
For the formal proof for this local property, please check our paper.
Please refer the paper for proof
Given a containing polygon
formed by points in the set

9. Localize Delaunay Triangulation
2018/11/17
Now what if we are only given an open polygon, which does not contain the given point inside.
In this case, if we still take the union of the circumcles of the triangles formed by the edges of the polygon and the given point,
We will miss some true Delaunay neighbors.
The light blue polygon shows the true Delaunay one-ring
However we can consider adding a virtual point, the infinity point, and make the polygon closed in some sense
Then we can add another two circumcircles, to be precisely, two half-planes
Now this new region defined by the circumcircles and the half-planes gives a localized region for the Delaunay neighbors of the red point
Given an open polygon? ∞

10. Half-plane intersecting for DT
Start from some initial (Delaunay) 1-ring
2018/11/17
Convergence of the alg.?
The approximate Voronoi polygon always shrinks
Now based on the local Delaunay property, we have a new half-plane intersecting algorithm for Delaunay triangulation
Suppose we have some initial 1-ring formed by candidates of Delaunay neighbors
Now for each triangle in the 1-ring, we check whether its circumcircle is empty,
if we find a point inside it, then we use it to run half-plane intersecting with the current approximate Voronoi polygon
We iteratively repeat this procedure for all the points in each of the circumcircle
In the end, all the circumcircle will be empty, and we get the true Delaunay one-ring
And also the Voronoi polygon
This algorithm is guaranteed to converge to the correct answer, as the approximate Vornoi polygon always shrinks

11. Parallel Delaunay Triangulation
2018/11/17
Now we are ready to present our parallel algorithm for Delaunay triangulation
Given a point set,
for each point in the set, we can find their Delaunay one-ring in parallel, using the half-plane intersecting algorithm
Merge all the Delaunay one-rings, we have the Delaunay triangulation
Find the Delaunay 1-ring for each point in parallel

12. Avoiding redundant computation
Each Delaunay triangle found 3 times
Each Delaunay edge found 4 times
2018/11/17
It is easy to see that by computing all the Delaunay one-ring independent of each other,
we will find each Delaunay triangle 3 times,
and each Delaunay edge 4 times
However actually for each point, we only need to find two Delaunay triangles incident to it
instead of all the one-ring Delaunay triangles
As long as for each point, the two corresponding Delaunay triangles pass through the vertical line through itself.
Of course, for point on the convex hull, we might find only one of them.
Based on this observation, we can avoid redundant computation in the half-plane intersecting algorithm.
Now we only need to iteratively do half-plane intersecting for the point inside the circumcirlce of the two triangles through the vertical
The algorithm converges for the same reason as the full algorithm.

13. Periodic DT
2018/11/17
Our algorithm can be used to compute Periodic Delaunay Triangluation
Here’s an example of Periodic Delauanay Triangulation

14. Periodic DT
2018/11/17
Actually for Periodic Delaunay triangulation, we can always find a containing polygon for each point.
In the worst case, we can always fall back to the polygon formed by it and replicas of itself in different periods

15. Results - Performance 2018/11/17
Here we show some results of our algorithm and its performance
Top left shows the runtime of the algorithm under different configurations.
The green curve corresponds to our algorithm with 4 cpu cores,
We can see it out performs all the other state-of-the-art sequential algorithms, including triangle, CGAL, qhull
In the bottom right, we show the speedup using multi-core CPU, we can see our algorithm achieves almost perfect speedup
In the bottom left and top right, we show different point distributions we use to compute the Delaunay triangulations,
Including both uniform and non-uniform distributions

16. Conclusion Local Delaunay triangulation Localize DT Parallel DT
Future work
GPU implementation
Optimization for non-uniform distribution
High dimensions
Manifold surfaces
2018/11/17
Now lets conclude this talk.
In this talk, we first introduce a local property of the Delaunay triangulation.
Then we use it to developed a parallel algorithm for computing the Delaunay triangulation based on half-plane intersecting
As for the future work, we want to implement the algorithm on the GPU,
Optimize the algorithm for non-uniform distributions
and generalize it to higher dimensions and non-Euclidean manifolds.

17. Thank you for your attention!
2018/11/17
Thank you for your attention!

18. Recovering regular DT from Periodic DT
2018/11/17