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Rectilinear Steiner Trees in Narrow Strips

Authors Henk Alkema, Mark de Berg

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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


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

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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)


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.

Subject Classification

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


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