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Touring a Sequence of Orthogonal Polygons

Authors Katrin Casel , Sándor Kisfaludi-Bak , Linda Kleist , Jeroen S.K. Lamme , Eunjin Oh , Yanheng Wang

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LIPIcs.ICALP.2026.50.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Dynamic programming
Keywords
  • shortest path
  • subquadratic time
  • dynamic planar distance oracle

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Abstract

We study the problem of computing a shortest tour that visits a sequence of k polygons P₁,…,P_k with a total number of n vertices. A tour is an oriented curve such that there exist points p_i ∈ P_i for all i where p_i appears not after p_{i+1}. In a seminal paper, Dror, Efrat, Lubiw and Mitchell (STOC 2003) considered the problem under L₂ distance, and gave Õ(nk) and Õ(nk²) algorithms for disjoint and intersecting convex polygons, respectively. In this paper, we consider the orthogonal setting (with orthogonal polygons and Manhattan distance) and obtain the following results:  
- a truly subquadratic Õ(n^{2-1/48}) algorithm when consecutive polygons in the sequence are disjoint; 
- an Õ(n) algorithm for ortho-convex polygons when consecutive polygons are disjoint; 
- an O(n) algorithm for axis-aligned rectangles; 
- Õ(n²) and Õ(n^{1.5}k²) algorithms without restrictions.  Our algorithms build on a wide range of techniques, including additively weighted Voronoi diagrams, rectangle decompositions, persistent data structures, and dynamic distance oracles for weighted planar graphs.

Cite As Get BibTex

Katrin Casel, Sándor Kisfaludi-Bak, Linda Kleist, Jeroen S.K. Lamme, Eunjin Oh, and Yanheng Wang. Touring a Sequence of Orthogonal Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 50:1-50:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.50

Author Details

Katrin Casel
  • Humboldt-University Berlin, Germany
Sándor Kisfaludi-Bak
  • Aalto University, Espoo, Finland
Linda Kleist
  • University of Hamburg, Germany
Jeroen S.K. Lamme
  • Eindhoven University of Technology, The Netherlands
Eunjin Oh
  • POSTECH, Pohang, South Korea
Yanheng Wang
  • ETH Zurich, Switzerland

Funding

  • Kisfaludi-Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.
  • Oh, Eunjin: Supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.RS-2024-00440239, Sublinear Scalable Algorithms for Large-Scale Data Analysis) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2024-00358505).
  • Wang, Yanheng: Part of this work was finished at Saarland University, Saarbrücken, Germany. It was part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).

Acknowledgements

This work was initiated at the 2025 Lorentz Center workshop "Fine-Grained and Parameterized Computational Geometry" in Leiden, the Netherlands.

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