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

On distance measures for polygonal curves bridging between Hausdorff and Fréchet distance

Hausdorff- und Fréchet-Abstand sind häufig genutzte Ähnlichkeitsmaße. Für Kurven, insbesondere Polygonzüge, können beide Distanzmaße effizient berechnet werden. In dieser Arbeit werden neue Ähnlichkeitsmaße ($\it Schnittdistanz$ und $\it Überdeckungsdistanz$, englisch $\textit {cut distance}$ und $\textit {cover distance}$) eingeführt, welche sowohl mit Fréchet- als auch mit Hausdorff-Abstand verwandt sind. Ein zusätzliches Ähnlichkeitsmaß ($\it Stationendistanz$, englisch $\textit {station distance}$) wird als Vereinfachung der Schnittdistanz vorgestellt. Das Entscheidungsproblem aller vorgestellten Ähnlichkeitsmaße ist NP-schwer, es werden aber algorithmische Ansätze wie Approximations- und FPT-Algorithmen vorgestellt. Das Free $\it Space-Diagramm$ ist das wichtigste Werkzeug zur Berechnung des Fréchet-Abstands, und das in dieser Arbeit untersuchte $\textit {Free Space-Realisierungsproblem}$ (englisch $\textit {free space realizability problem}$) stellt die Frage, ob eine gegebene Konfiguration mit dem Free Space-Diagramm eines Paars von Kurven übereinstimmt.Hausdorff and Fréchet distance are popular means to compare point sets, curves, and higher dimensional objects. We introduce the $\textit {cut distance}$ and the $\textit {cover distance}$, which bridge between Hausdorff and Fréchet distance. As a variant of the cut distance, we also introduce the $\textit {station distance}$. All novel distance measures are NP-hard to decide, but next to these reductions we present different algorithmic approaches such as approximation and FPT-algorithms for computing our distance measures. The free space and corresponding free space diagram is a key concept to efficiently computing the Fréchet distance and its variants. We study the free $\textit {free space realizability problem}$, i.e., we ask whether there exists a pair of curves such that their corresponding free space diagram resembles a given (potential) free space configuration. It is NP-hard to maximize the number of realized cells in a given diagram, but we also give algorithms that decide realizability for certain input instances

The k-Fréchet Distance: How to Walk Your Dog While Teleporting

We introduce a new distance measure for comparing polygonal chains: the k-Fréchet distance. As the name implies, it is closely related to the well-studied Fréchet distance but detects similarities between curves that resemble each other only piecewise. The parameter k denotes the number of subcurves into which we divide the input curves (thus we allow up to k-1 "teleports" on each input curve). The k-Fréchet distance provides a nice transition between (weak) Fréchet distance and Hausdorff distance. However, we show that deciding this distance measure turns out to be NP-hard, which is interesting since both (weak) Fréchet and Hausdorff distance are computable in polynomial time. Nevertheless, we give several possibilities to deal with the hardness of the k-Fréchet distance: besides a short exponential-time algorithm for the general case, we give a polynomial-time algorithm for k=2, i.e., we ask that we subdivide our input curves into two subcurves each. We can also approximate the optimal k by factor 2. We then present a more intricate FPT algorithm using parameters k (the number of allowed subcurves) and z (the number of segments of one curve that intersect the epsilon-neighborhood of a point on the other curve)

Across atoms to crossing continents: Application of similarity measures to biological location data

Biological processes involve movements across all measurable scales. Similarity measures can be applied to compare and analyze these movements but differ in how differences in movement are aggregated across space and time. The present study reviews frequently-used similarity measures, such as the Hausdorff distance, Fréchet distance, Dynamic Time Warping, and Longest Common Subsequence, jointly with several measures less used in biological applications (Wasserstein distance, weak Fréchet distance, and Kullback-Leibler divergence), and provides computational tools for each of them that may be used in computational biology. We illustrate the use of the selected similarity measures in diagnosing differences within two extremely contrasting sets of biological data, which, remarkably, may both be relevant for magnetic field perception by migratory birds. Specifically, we assess and discuss cryptochrome protein conformational dynamics and extreme migratory trajectories of songbirds between Alaska and Africa. We highlight how similarity measures contrast regarding computational complexity and discuss those which can be useful in noise elimination or, conversely, are sensitive to spatiotemporal scales

The k-Fréchet Distance: How to Walk Your Dog While Teleporting

We introduce a new distance measure for comparing polygonal chains: the k-Fréchet distance. As the name implies, it is closely related to the well-studied Fréchet distance but detects similarities between curves that resemble each other only piecewise. The parameter k denotes the number of subcurves into which we divide the input curves (thus we allow up to k-1 "teleports" on each input curve). The k-Fréchet distance provides a nice transition between (weak) Fréchet distance and Hausdorff distance. However, we show that deciding this distance measure turns out to be NP-hard, which is interesting since both (weak) Fréchet and Hausdorff distance are computable in polynomial time. Nevertheless, we give several possibilities to deal with the hardness of the k-Fréchet distance: besides a short exponential-time algorithm for the general case, we give a polynomial-time algorithm for k=2, i.e., we ask that we subdivide our input curves into two subcurves each. We can also approximate the optimal k by factor 2. We then present a more intricate FPT algorithm using parameters k (the number of allowed subcurves) and z (the number of segments of one curve that intersect the epsilon-neighborhood of a point on the other curve)