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.
@InProceedings{a.akitaya_et_al:LIPIcs.ISAAC.2023.3, author = {A. Akitaya, Hugo and Buchin, Maike and Mirzanezhad, Majid and Ryvkin, Leonie and Wenk, Carola}, title = {{Realizability of Free Spaces of Curves}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {3:1--3:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.3}, URN = {urn:nbn:de:0030-drops-193057}, doi = {10.4230/LIPIcs.ISAAC.2023.3}, annote = {Keywords: Fr\'{e}chet distance, Distance Geometry, free space diagram, inverse problem} }
Feedback for Dagstuhl Publishing