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Rectilinear Steiner Trees in Narrow Strips

Authors Henk Alkema, Mark de Berg

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • fixed-parameter tractable algorithms

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Abstract

A rectilinear Steiner tree for a set P of points in ℝ² is a tree that connects the points in P using horizontal and vertical line segments. The goal of {Minimum Rectilinear Steiner Tree} is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of {Minimum Rectilinear Steiner Tree} for point sets P inside the strip (-∞,+∞)× [0,δ] depends on the strip width δ. We obtain two main results.  
- We present an algorithm with running time n^O(√δ) for sparse point sets, that is, point sets where each 1×δ rectangle inside the strip contains O(1) points. 
- For random point sets, where the points are chosen randomly inside a rectangle of height δ and expected width n, we present an algorithm that is fixed-parameter tractable with respect to δ and linear in n. It has an expected running time of 2^{O(δ √{δ})} n.

Cite As Get BibTex

Henk Alkema and Mark de Berg. Rectilinear Steiner Trees in Narrow Strips. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.9

Author Details

Henk Alkema
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

The work in this paper is supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.

Acknowledgements

We thank Remco van der Hofstad for discussions about the probabilistic analysis.

References

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