The free space diagram is a popular tool to compute the well-known Fr\'echet
distance. As the Fr\'echet 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 Fr\'echet
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.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms
And Computations (ISAAC 2023