Computing the Fréchet Distance between Real-Valued Surfaces

Buchin, Kevin; Ophelders, Tim; Speckmann, Bettina

doi:10.1137/1.9781611974782.162

Cited by 7 publications

(15 citation statements)

References 21 publications

(28 reference statements)

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“…Let A, B ⊆ R d be two oriented manifolds and let f : A → M and g : B → M be two immersions. 6 Then the Fréchet distance between A and B is given by…”

Section: Fréchet Distancementioning

confidence: 99%

Distances Between Immersed Graphs: Metric Properties

et al. 2023

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Graphs in metric spaces appear in a wide range of data sets, and there is a large body of work focused on comparing, matching, or analyzing collections of graphs in different ambient spaces. In this survey, we provide an overview of a diverse collection of distance measures that can be defined on the set of finite graphs immersed (and in some cases, embedded) in a metric space. For each of the distance measures, we recall their definitions and investigate which of the properties of a metric they satisfy. Furthermore we compare the distance measures based on these properties and discuss their computational complexity.

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“…Let A, B ⊆ R d be two oriented manifolds and let f : A → M and g : B → M be two immersions. 6 Then the Fréchet distance between A and B is given by…”

Section: Fréchet Distancementioning

confidence: 99%

Distances Between Immersed Graphs: Metric Properties

et al. 2023

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“…There are various distances considered between graphs and trees in the literature such as "Graph edit distance" [6,13], and "Contour tree distance" [5]. The former does not respect the continuity of the curves and the latter is also a generalization of the Fréchet distance to graphs, but NP-hard to compute between them.…”

Section: Classification Of the Problemmentioning

confidence: 99%

Minimum-Complexity Graph Simplification under Fréchet-Like Distances

Filtser¹,

Mirzanezhad²,

Wenk³

2021

Preprint

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Simplifying graphs is a very applicable problem in numerous domains especially in computational geometry. Given a geometric graph and a threshold, the minimumcomplexity graph simplification asks for computing an alternative graph of minimum complexity so that the distance between the two graphs remains at most the threshold.In this paper we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph. When the input and output are trees, and the graph distance is applied from the simplified tree to the input tree, we give an O(kn 5 ) time algorithm, where k is the number of the leaves of the two trees that are identical and n is the number of vertices of the input.

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“…don.r.sheehy@gmail.com polygons [18], and other surfaces [20,5]. It should be noted that the general results for surfaces require exponential time.…”

Section: F (T) − G(h(t)) mentioning

confidence: 99%

“…It should be noted that the general results for surfaces require exponential time. Even for maps from R 2 → R, the problem was only just recently shown to be in NP and is NP-Hard to approximate within 2 − ε [5].…”

Section: F (T) − G(h(t)) mentioning

confidence: 99%

Fréchet-Stable Signatures Using Persistence Homology

Sheehy

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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For a metric space Y , the Fréchet distance is a metric on trajectories f, g :, g(h(t))) over continuous reparameterizations h of time. One can define the generalized Fréchet distance between more complex objects, functions f : X → Y where X is some topological space that minimizes over homeomorphisms from X → X. This more general definition has been studied for surfaces and often leads to computationally hard problems. We show how to compute in polynomial-time signatures for these functions for which the resulting metric on the signatures can also be computed in polynomial-time and provides a meaningful lower bound on the generalized Fréchet distance. Our approach uses persistent homology and exploits the natural invariance of persistence diagrams of functions to homeomorphisms of the domain. Our algorithm for computing the signatures in Euclidean spaces uses a new method for computing persistent homology of convex functions on simplicial complexes which may be of independent interest.

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Computing the Fréchet Distance between Real-Valued Surfaces

Cited by 7 publications

References 21 publications

Distances Between Immersed Graphs: Metric Properties

Distances Between Immersed Graphs: Metric Properties

Minimum-Complexity Graph Simplification under Fréchet-Like Distances

Fréchet-Stable Signatures Using Persistence Homology

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