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Kinetic locally minimal triangulation: theoretical evaluation and combinatorial analysis

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Abstract

Kinetic data structures represent an extension to ordinary data structures, where the underlying data become time dependent (e.g. moving points). In this paper, we define the kinetic locally minimal triangulation (KLMT) as a kinetic data structure extension to the locally minimal triangulation in the Euclidean plane. We explore the general properties of this data structure in order to show what types of events need to be considered during its life cycle; we also describe the predicates associated with these events. To describe the general kinetic features, we prove that KLMT is responsive, compact, efficient and non-local. In the combinatorial analysis of KLMT, we briefly describe the mathematical apparatus commonly used to investigate computational complexity properties of kinetic data structures and use it to establish the bounds on the number of events processed during the life cycle of this data structure. Finally, the obtained results are compared to the kinetic Delaunay triangulation showing that KLMT may provide some benefits over kinetic Delaunay triangulation, namely simplifying the mathematical equations that need to be computed in order to obtain the times of events.

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Acknowledgements

The authors would like to thank to RNDr. Svetlana Tomiczkova, Ph.D., from the Department of Mathematics of the Faculty of Applied Sciences at the University of West Bohemia for her invaluable help and proofreading of the text.

Funding

This work was funded by the Czech Science Foundation (Project Number 17-07690S) and by the Czech Ministry of Education, Youth and Sports (Project Number SGS-2019-016).

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  1. Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic

    Tomáš Vomáčka, Ivana Kolingerová & Martin Maňák

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  1. Tomáš Vomáčka
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  2. Ivana Kolingerová
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  3. Martin Maňák
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Correspondence to Tomáš Vomáčka.

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Vomáčka, T., Kolingerová, I. & Maňák, M. Kinetic locally minimal triangulation: theoretical evaluation and combinatorial analysis. Vis Comput 36, 757–765 (2020). https://doi.org/10.1007/s00371-019-01657-y

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