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Simple and Efficient Acceleration of the Smallest Enclosing Ball for Large Data Sets in \(E^2\): Analysis and Comparative Results

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Computational Science – ICCS 2022 (ICCS 2022)

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Abstract

Finding the smallest enclosing circle of the given points in \(E^2\) is a seemingly simple problem. However, already proposed algorithms have high memory requirements or require special solutions due to the great recursion depth or high computational complexity unacceptable for large data sets, etc. This paper presents a simple and efficient method with speed-up over 100 times based on processed data reduction. It is based on efficient preprocessing, which significantly reduces points used in the final processing. It also significantly reduces the depth of recursion and memory requirements, which is a limiting factor for large data processing. The proposed algorithm is easy to implement and it is extensible to the \(E^3\) case, too. The proposed algorithm was tested for up to \(10^9\) of points using the Halton’s and “Salt and Pepper” distributions.

Research supported by the University of West Bohemia - Institutional research support.

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Notes

  1. 1.

    Also, instead of computing the distance between points d, the \(\sqrt{d^2}\) should not be used and \(d^2\) can be used for distance comparisons Skala [52, 54]. Same idea can be applied to the radius of a circumscribed circle in Welzl’s algorithm.

  2. 2.

    Responsibilities: Algorithm design and analysis, manuscript preparation - V.Skala, Experimental implementation and verification - M. Cerny and J.Y. Saleh.

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Acknowledgment

The authors would like to thank to colleagues at the Shandong University in Jinan (China) and University of West Bohemia in Pilsen (Czech Rep.) for stimulating of this work and discussions made. Thanks belong also to the anonymous reviewers for their critical comments, hints provided and for unknown relevant references. \(^3\)(\(^3\)SIMD version using Intel’s intrinsics AVX-2 has been tested and led to an additional roughly 20% performance gain.)

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Authors and Affiliations

  1. Faculty of Applied Sciences, Department of Computer Science and Engineering, University of West Bohemia, Pilsen, CZ, 301 00, Czech Republic

    Vaclav Skala, Matej Cerny & Josef Yassin Saleh

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  1. Vaclav Skala
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  2. Matej Cerny
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  3. Josef Yassin Saleh
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Corresponding author

Correspondence to Vaclav Skala .

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Editors and Affiliations

  1. Brunel University London, London, UK

    Derek Groen

  2. University of Amsterdam, Amsterdam, The Netherlands

    Clélia de Mulatier

  3. AGH University of Science and Technology, Krakow, Poland

    Maciej Paszynski

  4. University of Amsterdam, Amsterdam, The Netherlands

    Valeria V. Krzhizhanovskaya

  5. University of Tennessee at Knoxville, Knoxville, TN, USA

    Jack J. Dongarra

  6. University of Amsterdam, Amsterdam, The Netherlands

    Peter M. A. Sloot

Appendix

Appendix

This appendix presents related papers to the Smallest Enclosing Ball problem:

Agarwal [1], Cavaleiro [2, 3], Cazals [4], Chen [5], Drager [6], Edelsbrunner [7], Efrat [8, 9], Elzinga [10], wiki: [70,71,72], Fischer [11,12,13], Friedman [14], Gaertner [15], Gao [16], Goaoc [17], Har-Peled [18], Jiang [19, 20], Kallberg [21], Karmakar [22, 23], Krivosija [24], Larsson [25], Li [26], Liu [27], Martinetz [28], Martyn [29], Matousek [30], Megiddo [31], Mordukhovich [32], Mukherjee [33], Munteanu [34], Nam [35], Nielsen [38,39,40], Nielsen [36, 37, 41, 42], Nock [43], Pan [44], Pronzato [45], Ritter [46], Saha [47], Shen [48], Shenmaier [49], Shi [50], Skyum [60], Smolik [61], Sylvester [62], Tao [63], Wang [64, 65], Wei [66], Welzl [67,68,69], Xu [73, 74], Yildirim [75], Zhou [76,77,78].

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Skala, V., Cerny, M., Saleh, J.Y. (2022). Simple and Efficient Acceleration of the Smallest Enclosing Ball for Large Data Sets in \(E^2\): Analysis and Comparative Results. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13350. Springer, Cham. https://doi.org/10.1007/978-3-031-08751-6_52

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