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An Output-Sensitive Algorithm for Computing the Union of Cubes and Fat Boxes in 3D

Authors Pankaj K. Agarwal, Alex Steiger

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Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Alex Steiger
  • Department of Computer Science, Duke University, Durham, NC, USA

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Pankaj K. Agarwal and Alex Steiger. An Output-Sensitive Algorithm for Computing the Union of Cubes and Fat Boxes in 3D. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 10:1-10:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Let C be a set of n axis-aligned cubes of arbitrary sizes in ℝ³. Let U be their union, and let κ be the number of vertices on ∂U; κ can vary between O(1) and O(n²). We show that U can be computed in O(n log³ n + κ) time if C is in general position. The algorithm also computes the union of a set of fat boxes (i.e., boxes with bounded aspect ratio) within the same time bound. If the cubes in C are congruent or have bounded depth, the running time improves to O(n log² n), and if both conditions hold, the running time improves to O(n log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • union of cubes
  • fat boxes
  • plane-sweep


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