Realizability of Free Spaces of Curves

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 $\mathbb{R}^{> 2}$, showing $\exists\mathbb{R}$-hardness. We use this to show that for curves in $\mathbb{R}^{\ge 2}$, the realizability problem is $\exists\mathbb{R}$-complete, both for continuous and for discrete Fr\'echet distance. We prove that the continuous case in $\mathbb{R}^1$ 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 $\mathbb{R}^1$, 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

FRESH: Fréchet similarity with hashing

This paper studies the r-range search problem for curves under the continuous Fréchet distance: given a dataset S of n polygonal curves and a threshold >0 , construct a data structure that, for any query curve q, efficiently returns all entries in S with distance at most r from q. We propose FRESH, an approximate and randomized approach for r-range search, that leverages on a locality sensitive hashing scheme for detecting candidate near neighbors of the query curve, and on a subsequent pruning step based on a cascade of curve simplifications. We experimentally compare FRESH to exact and deterministic solutions, and we show that high performance can be reached by suitably relaxing precision and recall