LIPIcs.GD.2024.3.pdf
- Filesize: 0.86 MB
- 23 pages
Document
The Euclidean MST-Ratio for Bi-Colored Lattices
Authors
Sebastiano Cultrera di Montesano 
,
Ondřej Draganov ,
Herbert Edelsbrunner ,
Morteza Saghafian
-
Part of:
Volume:
32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Graph Drawing and Network Visualization (GD) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-10-28
File
Document Identifiers
Author Details
- Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Cite AsGet 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
https://doi.org/10.4230/LIPIcs.GD.2024.3
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.
Subject Classification
ACM Subject Classification
- Theory of computation → Computational geometry
Keywords
- Minimum spanning Trees
- Steiner Ratio
- Lattices
- Partitions
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Gunnar E. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46:255-308, 2009. URL: https://doi.org/10.1090/S0273-0979-09-01249-X.
- F. Chung and R. Graham. A new bound for Euclidean Steiner minimal trees. Annals of the New York Academy of Sciences, 440:328-346, 1985. URL: https://doi.org/10.1111/j.1749-6632.1985.tb14564.x.
- David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete Comput. Geom., pages 103-120, 2007. URL: https://doi.org/10.1007/s00454-006-1276-5.
- David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Persistent homology for kernels, images, and cokernels. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1011-1020, January 2009. URL: https://doi.org/10.1137/1.9781611973068.110.
- Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. Chromatic alpha complexes, 2024. URL: https://doi.org/10.48550/arXiv.2212.03128.
- Adrian Dumitrescu, János Pach, and Géza Tóth. Two trees are better than one, 2023. URL: https://doi.org/10.48550/arXiv.2312.09916.
- Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. Springer, January 2010. URL: https://doi.org/10.1007/978-3-540-33259-6_7.
- E. N. Gilbert and H. O. Pollak. Steiner minimal trees. SIAM J. Appl. Math., 16(1):1-29, January 1968. URL: https://doi.org/10.1137/0116001.
- V. Jarník. O jistém problému minimálním: (z dopisu panu O. Borůvkovi). Mor. přírodovědecká společnost, 1930. URL: http://dml.cz/dmlcz/500726.
- V. Jarník and M. Kössler. O minimálních grafech, obsahujících n daných bodů. Časopis pro pěstování matematiky a fysiky, 063(8):223-235, 1934. URL: https://doi.org/10.21136/CPMF.1934.122548.
- Joseph B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7(1):48-50, 1956. URL: https://doi.org/10.2307/2033241.
- B. Zhilinskii. Introduction to Louis Michel’s Lattice Geometry Through Group Action. Current natural sciences. EDP Sciences, 2015. URL: http://library.oapen.org/handle/20.500.12657/25012.