Realizability of Free Spaces of Curves

Authors Hugo A. Akitaya , Maike Buchin , Majid Mirzanezhad , Leonie Ryvkin , Carola Wenk

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Hugo A. Akitaya
  • Department of Computer Science, University of Massachusetts Lowell, MA, USA
Maike Buchin
  • Department of Computer Science, Ruhr University Bochum, Germany
Majid Mirzanezhad
  • Transportation Research Institute, University of Michigan, Ann Arbor, MI, USA
Leonie Ryvkin
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
Carola Wenk
  • Department of Computer Science, Tulane University, New Orleans, LA, USA

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Hugo A. Akitaya, Maike Buchin, Majid Mirzanezhad, Leonie Ryvkin, and Carola Wenk. Realizability of Free Spaces of Curves. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is realizable, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in ℝ^{>2}, showing ∃ℝ-hardness. We use this to show that for curves in ℝ^{≥2} the realizability problem is ∃ℝ-complete, both for continuous and for discrete Fréchet distance. We prove that the continuous case in ℝ¹ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in ℝ¹ we show that the problem becomes solvable in polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Fréchet distance
  • Distance Geometry
  • free space diagram
  • inverse problem


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