Skip to main content

Load-Balancing for Parallel Delaunay Triangulations

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11725)


Computing the Delaunay triangulation (DT) of a given point set in \(\mathbb {R}^D\) is one of the fundamental operations in computational geometry. Recently, Funke and Sanders [11] presented a divide-and-conquer DT algorithm that merges two partial triangulations by re-triangulating a small subset of their vertices – the border vertices – and combining the three triangulations efficiently via parallel hash table lookups. The input point division should therefore yield roughly equal-sized partitions for good load-balancing and also result in a small number of border vertices for fast merging. In this paper, we present a novel divide-step based on partitioning the triangulation of a small sample of the input points. In experiments on synthetic and real-world data sets, we achieve nearly perfectly balanced partitions and small border triangulations. This almost cuts running time in half compared to non-data-sensitive division schemes on inputs exhibiting an exploitable underlying structure.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-29400-7_12
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-29400-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.


  1. 1.

    For a more detailed description of the primitives we refer to the technical report [12].

  2. 2.

  3. 3.


  4. 4.

    \(\omega (e = (v,w)) = -\log d(v,w)\) width \(d(\cdot )\) denoting the Euclidean distance.


  1. Aggarwal, A., Chazelle, B., Guibas, L.: Parallel computational geometry. Algorithmica 3(1), 293–327 (1988)

    MathSciNet  CrossRef  Google Scholar 

  2. Akhremtsev, Y., Sanders, P., Schulz, C.: High-quality shared-memory graph partitioning. In: Aldinucci, M., Padovani, L., Torquati, M. (eds.) Euro-Par 2018. LNCS, vol. 11014, pp. 659–671. Springer, Cham (2018).

    CrossRef  Google Scholar 

  3. Batista, V.H., Millman, D.L., Pion, S., Singler, J.: Parallel geometric algorithms for multi-core computers. Comp. Geom. 43(8), 663–677 (2010)

    MathSciNet  CrossRef  Google Scholar 

  4. van den Bergen, G.: Efficient collision detection of complex deformable models using aabb trees. J. Graph. Tools 2(4), 1–13 (1997)

    CrossRef  Google Scholar 

  5. Chen, M.B.: The merge phase of parallel divide-and-conquer scheme for 3D Delaunay triangulation. In: International Symposium on Parallel and Distributed Processing with Applications (ISPA), pp. 224–230, IEEE (2010)

    Google Scholar 

  6. Chrisochoides, N.: Parallel mesh generation. Numerical Solution of Partial Differential Equations on Parallel Computers. LNCS, vol. 51, pp. 237–264. Springer, Berlin (2006).

    CrossRef  MATH  Google Scholar 

  7. Chrisochoides, N., Nave, D.: Simultaneous mesh generation and partitioning for Delaunay meshes. Math. Comput. Sim. 54(4), 321–339 (2000)

    MathSciNet  CrossRef  Google Scholar 

  8. Cignoni, P., Montani, C., Scopigno, R.: DeWall: a fast divide and conquer Delaunay triangulation algorithm in \(E^d\). CAD 30(5), 333–341 (1998)

    MATH  Google Scholar 

  9. Collaboration, G.: Gaia data release 2. summary of the contents and survey properties. arXiv (abs/1804.09365) (2018)

    Google Scholar 

  10. Devillers, O.: The Delaunay hierarchy. Int. J. Found. Comput. Sci. 13(02), 163–180 (2002)

    MathSciNet  CrossRef  Google Scholar 

  11. Funke, D., Sanders, P.: Parallel \(d\)-d Delaunay triangulations in shared and distributed memory. In: ALENEX, pp. 207–217, SIAM (2017)

    Google Scholar 

  12. Funke, D., Sanders, P., Winkler, V.: Load-Balancing for Parallel Delaunay Triangulations. arXiv (abs/1902.07554) (2019)

    Google Scholar 

  13. Hert, S., Seel, M.: dD convex hulls and delaunay triangulations. In: CGAL User and Reference Manual, CGAL Editorial Board, 4.7 edn. (2015)

    Google Scholar 

  14. Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Techn. J. 49(2), 291–307 (1970)

    CrossRef  Google Scholar 

  15. Kohout, J., Kolingerová, I., Žára, J.: Parallel Delaunay triangulation in E2 and E3 for computers with shared memory. Par. Comp. 31(5), 491–522 (2005)

    CrossRef  Google Scholar 

  16. Larsson, T., Akenine-Möller, T., Lengyel, E.: On faster sphere-box overlap testing. J. Graph., GPU, Game Tools 12(1), 3–8 (2007)

    CrossRef  Google Scholar 

  17. Lee, S., Park, C.I., Park, C.M.: An improved parallel algorithm for Delaunay triangulation on distributed memory parallel computers. Parallel Process. Lett. 11, 341–352 (2001)

    CrossRef  Google Scholar 

  18. Sanders, P., Lamm, S., Hübschle-Schneider, L., Schrade, E., Dachsbacher, C.: Efficient parallel random sampling - vectorized, cache-efficient, and online. ACM Trans. Math. Softw. 44(3), 29:1–29:14 (2018)

    MathSciNet  CrossRef  Google Scholar 

  19. Sanders, P., Schulz, C.: Think locally, act globally: highly balanced graph partitioning. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 164–175. Springer, Heidelberg (2013).

    CrossRef  Google Scholar 

  20. Shewchuk, J.: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. Appl. Comp. Geom. Towards Geom. Eng. 1148, 203–222 (1996)

    CrossRef  Google Scholar 

  21. Shewchuk, J.: Adaptive precision floating-point arithmetic and fast robust geometric predicates. Disc. Comp. Geom. 18(3), 305–363 (1997)

    MathSciNet  CrossRef  Google Scholar 

  22. Simon, H.D., Teng, S.H.: How good is recursive bisection? J. Sci. Comput. 18(5), 1436–1445 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Su, P., Drysdale, R.L.S.: A comparison of sequential delaunay triangulation algorithms. In: Symposium on Computing Geometry (SCG), pp. 61–70, ACM (1995)

    Google Scholar 

Download references


The authors gratefully acknowledge the Deutsche Forschungsgemeinschaft (DFG) who partially supported this work under grants SA 933/10-2 and SA 933/11-1.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniel Funke .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Funke, D., Sanders, P., Winkler, V. (2019). Load-Balancing for Parallel Delaunay Triangulations. In: Yahyapour, R. (eds) Euro-Par 2019: Parallel Processing. Euro-Par 2019. Lecture Notes in Computer Science(), vol 11725. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29399-4

  • Online ISBN: 978-3-030-29400-7

  • eBook Packages: Computer ScienceComputer Science (R0)