For a hypergraph ℋ = (X,ℰ) a support is a graph G on X such that for each E ∈ ℰ, the induced subgraph of G on the elements in E is connected. If G is planar, we call it a planar support. A set of axis parallel rectangles ℛ forms a non-piercing family if for any R₁, R₂ ∈ ℛ, R₁⧵R₂ is connected. Given a set P of n points in ℝ² and a set ℛ of m non-piercing axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph (P,ℛ) in O(nlog² n + (n+m)log m) time, where each R ∈ ℛ defines a hyperedge consisting of all points of P contained in R.
@InProceedings{pal_et_al:LIPIcs.ISAAC.2024.53, author = {Pal, Ambar and Raman, Rajiv and Ray, Saurabh and Singh, Karamjeet}, title = {{A Fast Algorithm for Computing a Planar Support for Non-Piercing Rectangles}}, booktitle = {35th International Symposium on Algorithms and Computation (ISAAC 2024)}, pages = {53:1--53:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-354-6}, ISSN = {1868-8969}, year = {2024}, volume = {322}, editor = {Mestre, Juli\'{a}n and Wirth, Anthony}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.53}, URN = {urn:nbn:de:0030-drops-221819}, doi = {10.4230/LIPIcs.ISAAC.2024.53}, annote = {Keywords: Algorithms, Hypergraphs, Computational Geometry, Visualization} }
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