An External Memory Algorithm for the Minimum Enclosing Ball Problem

Linus Källberg, Evan Shellshear, Thomas Larsson

2016

Abstract

In this article we present an external memory algorithm for computing the exact minimum enclosing ball of a massive set of points in any dimension. We test the performance of the algorithm on real-life three-dimensional data sets and demonstrate for the first time the practical efficiency of exact out-of-core algorithms. By use of simple heuristics, we achieve near-optimal I/O in all our test cases.

References

  1. Agarwal, P. K. and Sharathkumar, R. (2010). Streaming algorithms for extent problems in high dimensions. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1481- 1489. Society for Industrial and Applied Mathematics.
  2. Aggarwal, A., Vitter, J., et al. (1988). The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116-1127.
  3. Ajwani, D., Beckmann, A., Jacob, R., Meyer, U., and Moruz, G. (2009). On computational models for flash memory devices. In Experimental Algorithms, pages 16-27. Springer.
  4. ATTO Technology, Inc. (2010). ATTO Disk Benchmark v2.47. http://www.attotech.com/disk-benchmark/.
  5. Ba?doiu, M. and Clarkson, K. L. (2008). Optimal Core-Sets for Balls. Computational Geometry, 40(1):14-22.
  6. Ba?doiu, M., Har-Peled, S., and Indyk, P. (2002). Approximate clustering via core-sets. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing (STOC 7802), page 250.
  7. Borrmann, D., Elseberg, J., Houshiar, H., and N üchter, A. (2011). Tower data set. http://kos.informatik.uniosnabrueck.de/3Dscans/. Jacobs University Bremen.
  8. Breimann, C. and Vahrenhold, J. (2003). External memory computational geometry revisited. In Meyer, U., Sanders, P., and Sibeyn, J., editors, Algorithms for Memory Hierarchies, volume 2625 of Lecture Notes in Computer Science, pages 110-148. Springer.
  9. Chan, T. M. and Pathak, V. (2014). Streaming and dynamic algorithms for minimum enclosing balls in high dimensions. Computational Geometry, 47(2):240-247.
  10. Fischer, K., Gärtner, B., and Kutz, M. (2003). Fast Smallest-Enclosing-Ball Computation in High Dimensions. In Proceedings of the 11th European Symposium on Algorithms (ESA 7803), pages 630-641.
  11. Gärtner, B. (1999). Fast and robust smallest enclosing balls. In Proceedings of the 7th Annual European Symposium on Algorithms (ESA 7899), pages 325-338. Springer.
  12. IDC (2014). The Digital Universe of Opportunities: Rich Data and the Increasing Value of the Internet of Things. http://www.emc.com/leadership/digitaluniverse/2014iview/.
  13. Källberg, L. and Larsson, T. (2013). Faster approximation of minimum enclosing balls by distance filtering and GPU parallelization. Journal of Graphics Tools, 17(3):67-84.
  14. Kumar, P., Mitchell, J. S. B., and Yildirim, E. A. (2003). Approximate minimum enclosing balls in high dimensions using core-sets. Journal of Experimental Algorithmics, 8.
  15. Larsson, T. and Källberg, L. (2013). Fast and robust approximation of smallest enclosing balls in arbitrary dimensions. Computer Graphics Forum, 32(5):93-101.
  16. Nielsen, F. and Nock, R. (2009). Approximating smallest enclosing balls with applications to machine learning. International Journal of Computational Geometry & Applications, 19(5).
  17. Stanford Graphics Laboratory (2015). The Digital Michelangelo Project Archive of 3D Models. http://graphics.stanford.edu/data/dmich-public/.
  18. Welzl, E. (1991). Smallest enclosing disks (balls and ellipsoids). New results and new trends in computer science, 555(3075):359-370.
  19. Yildirim, E. A. (2008). Two algorithms for the minimum enclosing ball problem. SIAM Journal on Optimization, 19(3):1368-1391.
  20. Zarrabi-Zadeh, H. and Chan, T. M. (2006). A simple streaming algorithm for minimum enclosing balls. In Proceedings of the 18th Annual Canadian Conference on Computational Geometry (CCCG 7806).
Download


Paper Citation


in Harvard Style

Källberg L., Shellshear E. and Larsson T. (2016). An External Memory Algorithm for the Minimum Enclosing Ball Problem . In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016) ISBN 978-989-758-175-5, pages 83-90. DOI: 10.5220/0005675600810088


in Bibtex Style

@conference{grapp16,
author={Linus Källberg and Evan Shellshear and Thomas Larsson},
title={An External Memory Algorithm for the Minimum Enclosing Ball Problem},
booktitle={Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016)},
year={2016},
pages={83-90},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005675600810088},
isbn={978-989-758-175-5},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016)
TI - An External Memory Algorithm for the Minimum Enclosing Ball Problem
SN - 978-989-758-175-5
AU - Källberg L.
AU - Shellshear E.
AU - Larsson T.
PY - 2016
SP - 83
EP - 90
DO - 10.5220/0005675600810088