# Efficient Delaunay Tessellation through K-D Tree Decomposition

## Abstract

Delaunay tessellations are fundamental data structures in computational geometry. They are important in data analysis, where they can represent the geometry of a point set or approximate its density. The algorithms for computing these tessellations at scale perform poorly when the input data is unbalanced. We investigate the use of k-d trees to evenly distribute points among processes and compare two strategies for picking split points between domain regions. Because resulting point distributions no longer satisfy the assumptions of existing parallel Delaunay algorithms, we develop a new parallel algorithm that adapts to its input and prove its correctness. We evaluate the new algorithm using two late-stage cosmology datasets. The new running times are up to 50 times faster using k-d tree compared with regular grid decomposition. Moreover, in the unbalanced data sets, decomposing the domain into a k-d tree is up to five times faster than decomposing it into a regular grid.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- Computational Research Division

- OSTI Identifier:
- 1375632

- Report Number(s):
- LBNL-1007265

ir:1007265

- Resource Type:
- Conference

- Resource Relation:
- Conference: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, Salt Lake City, Utah, 11/13/2016

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Morozov, Dmitriy, and Peterka, Tom.
```*Efficient Delaunay Tessellation through K-D Tree Decomposition*. United States: N. p., 2017.
Web. doi:10.1109/SC.2016.61.

```
Morozov, Dmitriy, & Peterka, Tom.
```*Efficient Delaunay Tessellation through K-D Tree Decomposition*. United States. doi:10.1109/SC.2016.61.

```
Morozov, Dmitriy, and Peterka, Tom. Mon .
"Efficient Delaunay Tessellation through K-D Tree Decomposition". United States. doi:10.1109/SC.2016.61. https://www.osti.gov/servlets/purl/1375632.
```

```
@article{osti_1375632,
```

title = {Efficient Delaunay Tessellation through K-D Tree Decomposition},

author = {Morozov, Dmitriy and Peterka, Tom},

abstractNote = {Delaunay tessellations are fundamental data structures in computational geometry. They are important in data analysis, where they can represent the geometry of a point set or approximate its density. The algorithms for computing these tessellations at scale perform poorly when the input data is unbalanced. We investigate the use of k-d trees to evenly distribute points among processes and compare two strategies for picking split points between domain regions. Because resulting point distributions no longer satisfy the assumptions of existing parallel Delaunay algorithms, we develop a new parallel algorithm that adapts to its input and prove its correctness. We evaluate the new algorithm using two late-stage cosmology datasets. The new running times are up to 50 times faster using k-d tree compared with regular grid decomposition. Moreover, in the unbalanced data sets, decomposing the domain into a k-d tree is up to five times faster than decomposing it into a regular grid.},

doi = {10.1109/SC.2016.61},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2017},

month = {8}

}