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1. Fast Fréchet Distance for Curves with Long Edges

   Gudmundsson, J.; Mirzanezhad, M.; Mohades, A.; Wenk, C. (April 2018, 3rd International Workshop on Interactive and Spatial Computing)

   Computing Fréchet distance between two curves takes roughly quadratic time. The only strongly subquadratic time algorithm has been proposed in [7] for c-packed curves. In this paper, we show that for curves with long edges the Fréchet distance computations become easier. Let P and Q be two polygonal curves in Rd with n and m vertices, respectively. We prove three main results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, and (3) a linear-time [EQUATION]-approximation algorithm for approximating the Fréchet distance between two curves. 

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2. Fréchet Edit Distance

   Fox, Emily; Nayyeri, Amir; Perry, Jonathan James; Raichel, Benjmain (June 2024, Proceedings of the 40th International Symposium on Computational Geometry)

   In this paper we define and investigate the Fréchet edit distance problem. Here, given two polygonal curves $$\pi$$ and $$\sigma$$ and a threshhold value $$\delta$$ , we seek the minimum number of edits to $$\sigma$$ such that the Fréchet distance between the edited curve and $$\pi$$ is at most $$\delta$$. For the edit operations we consider three cases, namely, deletion of vertices, insertion of vertices, or both. For this basic problem we consider a number of variants. Specifically, we provide polynomial time algorithms for both discrete and continuous Fréchet edit distance variants, as well as hardness results for weak Fréchet edit distance variants. 

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3. Fréchet Edit Distance

   https://doi.org/10.4230/LIPIcs.SoCG.2024.58  

   Fox, Emily; Nayyeri, Amir; Perry, Jonathan James; Raichel, Benjamin (June 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   We define and investigate the Fréchet edit distance problem. Given two polygonal curves π and σ and a non-negative threshhold value δ, we seek the minimum number of edits to σ such that the Fréchet distance between the edited σ and π is at most δ. For the edit operations we consider three cases, namely, deletion of vertices, insertion of vertices, or both. For this basic problem we consider a number of variants. Specifically, we provide polynomial time algorithms for both discrete and continuous Fréchet edit distance variants, as well as hardness results for weak Fréchet edit distance variants. 

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4. Discrete Fréchet Distance Oracles

   https://doi.org/10.4230/LIPIcs.SoCG.2024.10  

   Aronov, Boris; Farhana, Tsuri; Katz, Matthew J; Ramesh, Indu (June 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   It is unlikely that the discrete Fréchet distance between two curves of length n can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, P, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to P in sublinear time. Since there is evidence that this is impossible for query curves of length Θ(n^α), for any α > 0, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds. We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, t-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph G in the family, so that, given a query segment and a pair u,v of vertices in G, one can quickly compute the smallest discrete Fréchet distance between the segment and any (u,v)-path in G. The answer is exact, if t = 1, and approximate if t > 1. 

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5. Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves

   Brüning, F.; Akitaya, H.; Chambers, E; Driemel, A (February 2023, Computing in geometry and topology)

   We study subtrajectory clustering under the Fréchet distance. Given one or more trajectories, the task is to split the trajectories into several parts, such that the parts have a good clustering structure. We approach this problem via a new set cover formulation, which we think provides a natural formalization of the problem as it is studied in many applications. Given a polygonal curve P with n vertices in fixed dimension, integers k, ℓ ≥ 1, and a real value Δ > 0, the goal is to find k center curves of complexity at most ℓ such that every point on P is covered by a subtrajectory that has small Fréchet distance to one of the k center curves (≤ Δ). In many application scenarios, one is interested in finding clusters of small complexity, which is controlled by the parameter ℓ. Our main result is a bicriterial approximation algorithm: if there exists a solution for given parameters k, ℓ, and Δ, then our algorithm finds a set of k' center curves of complexity at most ℓ with covering radius Δ' with k' in O(kℓ2 log (kℓ)), and Δ' ≤ 19Δ. Moreover, within these approximation bounds, we can minimize k while keeping the other parameters fixed. If ℓ is a constant independent of n, then, the approximation factor for the number of clusters k is O(log k) and the approximation factor for the radius Δ is constant. In this case, the algorithm has expected running time in Õ(km2 + mn) and uses space in O(n + m), where m=⌈L/Δ⌉ and L is the total arclength of the curve P. 

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