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An \(\varOmega (n^3)\) Lower Bound on the Number of Cell Crossings for Weighted Shortest Paths in 3-Dimensional Polyhedral Structures

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LATIN 2020: Theoretical Informatics (LATIN 2021)

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Abstract

A new lower bound of \(\varOmega (n^3)\) on the maximum number of cell crossings for weighted shortest paths in 3-dimensional polyhedral structures consisting of a linear number of \(\mathcal {O}(n)\) polyhedral cells and cell faces is derived. This is a generalization and sharpening of the formerly known \(\varOmega (n^2)\) lower bound on the maximum number of cell crossings for weighted shortest path in 2-dimensional polyhedral structures and has been a long-standing open problem for the 3-dimensional case.

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Acknowledgement

The authors would like to thank the referees for valuable comments made. The third author would like to thank Erik van Leeuwen, Ioana Bercea, Karl Bringmann, and Michael Sagraloff for fruitful initial discussions which took place while that author was visiting Max-Planck Institute, Algorithms, Saarbrücken.

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Authors and Affiliations

  1. Hochschule für Technik und Wirtschaft, Computer Engineering, Berlin, Germany

    Frank Bauernöppel

  2. Carleton University, School of Computer Science, Ottawa, Canada

    Anil Maheshwari & Jörg-Rüdiger Sack

Authors
  1. Frank Bauernöppel
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  2. Anil Maheshwari
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  3. Jörg-Rüdiger Sack
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Corresponding author

Correspondence to Frank Bauernöppel .

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Editors and Affiliations

  1. University of São Paulo, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa

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Bauernöppel, F., Maheshwari, A., Sack, JR. (2020). An \(\varOmega (n^3)\) Lower Bound on the Number of Cell Crossings for Weighted Shortest Paths in 3-Dimensional Polyhedral Structures. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_19

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  • DOI: https://doi.org/10.1007/978-3-030-61792-9_19

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-61792-9

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