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The Euclidean MST-Ratio for Bi-Colored Lattices

Authors Sebastiano Cultrera di Montesano , Ondřej Draganov , Herbert Edelsbrunner , Morteza Saghafian

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Minimum spanning Trees
  • Steiner Ratio
  • Lattices
  • Partitions

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Abstract

Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25.

Cite As Get BibTex

Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. The Euclidean MST-Ratio for Bi-Colored Lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.3

Author Details

Sebastiano Cultrera di Montesano
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Ondřej Draganov
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Herbert Edelsbrunner
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Morteza Saghafian
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics", Austrian Science Fund (FWF), grant no. I 02979-N35.

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