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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen–Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.

Ulrich Bauer and Fabian Roll. Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2024.15, author = {Bauer, Ulrich and Roll, Fabian}, title = {{Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.15}, URN = {urn:nbn:de:0030-drops-199600}, doi = {10.4230/LIPIcs.SoCG.2024.15}, annote = {Keywords: persistent homology, discrete Morse theory, apparent pairs, Wrap complex, lexicographic optimal chains, shape reconstruction} }

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**Published in:** Dagstuhl Reports, Volume 13, Issue 5 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 23192 "Topological Data Analysis and Applications". The seminar brought together researchers with backgrounds in mathematics, computer science, and different application domains with the aim of identifying and exploring emerging directions within computational topology for data analysis. This seminar was designed to be a followup event to two successful Dagstuhl Seminars (17292, July 2017; 19212, May 2019). The list of topics and participants were updated to reflect recent developments and to engage wider participation. Close interaction between the participants during the seminar accelerated the convergence between mathematical and computational thinking in the development of theories and scalable algorithms for data analysis, and the identification of different applications of topological analysis.

Ulrich Bauer, Vijay Natarajan, and Bei Wang. Topological Data Analysis and Applications (Dagstuhl Seminar 23192). In Dagstuhl Reports, Volume 13, Issue 5, pp. 71-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{bauer_et_al:DagRep.13.5.71, author = {Bauer, Ulrich and Natarajan, Vijay and Wang, Bei}, title = {{Topological Data Analysis and Applications (Dagstuhl Seminar 23192)}}, pages = {71--95}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {5}, editor = {Bauer, Ulrich and Natarajan, Vijay and Wang, Bei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.5.71}, URN = {urn:nbn:de:0030-drops-193652}, doi = {10.4230/DagRep.13.5.71}, annote = {Keywords: algorithms, applications, computational topology, topological data analysis, visualization} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We present an algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. The algorithm makes use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. The clearing optimization works particularly well in the context of relative cohomology, and using previous duality results we can translate the barcodes of images in relative cohomology to those in absolute homology. This forms the basis for an implementation of image persistence computations for inclusions of filtrations of Vietoris-Rips complexes in the framework of the software Ripser.

Ulrich Bauer and Maximilian Schmahl. Efficient Computation of Image Persistence. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2023.14, author = {Bauer, Ulrich and Schmahl, Maximilian}, title = {{Efficient Computation of Image Persistence}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.14}, URN = {urn:nbn:de:0030-drops-178643}, doi = {10.4230/LIPIcs.SoCG.2023.14}, annote = {Keywords: Persistent homology, image persistence, barcode computation} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Clearing is a simple but effective optimization for the standard algorithm of persistent homology (ph), which dramatically improves the speed and scalability of ph computations for Vietoris-Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris-Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter ph.
We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter ph that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.

Ulrich Bauer, Fabian Lenzen, and Michael Lesnick. Efficient Two-Parameter Persistence Computation via Cohomology. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2023.15, author = {Bauer, Ulrich and Lenzen, Fabian and Lesnick, Michael}, title = {{Efficient Two-Parameter Persistence Computation via Cohomology}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.15}, URN = {urn:nbn:de:0030-drops-178656}, doi = {10.4230/LIPIcs.SoCG.2023.15}, annote = {Keywords: Persistent homology, persistent cohomology, two-parameter persistence, clearing} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set 𝒮 of r-dimensional simplices of minimum cardinality so that 𝒮 meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+Δ, where Δ is the maximum degree of the Hasse graph of the complex 𝖪.

Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi. On Computing Homological Hitting Sets. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.ITCS.2023.13, author = {Bauer, Ulrich and Rathod, Abhishek and Zehavi, Meirav}, title = {{On Computing Homological Hitting Sets}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {13:1--13:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.13}, URN = {urn:nbn:de:0030-drops-175169}, doi = {10.4230/LIPIcs.ITCS.2023.13}, annote = {Keywords: Algorithmic topology, Cut problems, Surfaces, Parameterized complexity} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We establish bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.

Ulrich Bauer, Håvard Bakke Bjerkevik, and Benedikt Fluhr. Quasi-Universality of Reeb Graph Distances. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2022.14, author = {Bauer, Ulrich and Bjerkevik, H\r{a}vard Bakke and Fluhr, Benedikt}, title = {{Quasi-Universality of Reeb Graph Distances}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {14:1--14:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.14}, URN = {urn:nbn:de:0030-drops-160221}, doi = {10.4230/LIPIcs.SoCG.2022.14}, annote = {Keywords: Reeb graphs, contour trees, merge trees, distances, universality, interleaving distance, functional distortion distance, functional contortion distance} }

Document

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Motivated by computational aspects of persistent homology for Vietoris–Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris–Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris–Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data.

Ulrich Bauer and Fabian Roll. Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2022.15, author = {Bauer, Ulrich and Roll, Fabian}, title = {{Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.15}, URN = {urn:nbn:de:0030-drops-160237}, doi = {10.4230/LIPIcs.SoCG.2022.15}, annote = {Keywords: Vietoris–Rips complexes, persistent homology, discrete Morse theory, apparent pairs, hyperbolicity, geodesic defect, Ripser} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

Ulrich Bauer, Claudia Landi, and Facundo Mémoli. The Reeb Graph Edit Distance Is Universal. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2020.15, author = {Bauer, Ulrich and Landi, Claudia and M\'{e}moli, Facundo}, title = {{The Reeb Graph Edit Distance Is Universal}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.15}, URN = {urn:nbn:de:0030-drops-121730}, doi = {10.4230/LIPIcs.SoCG.2020.15}, annote = {Keywords: Reeb graphs, topological descriptors, edit distance, interleaving distance} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.

Ulrich Bauer, Abhishek Rathod, and Jonathan Spreer. Parametrized Complexity of Expansion Height. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bauer_et_al:LIPIcs.ESA.2019.13, author = {Bauer, Ulrich and Rathod, Abhishek and Spreer, Jonathan}, title = {{Parametrized Complexity of Expansion Height}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.13}, URN = {urn:nbn:de:0030-drops-111346}, doi = {10.4230/LIPIcs.ESA.2019.13}, annote = {Keywords: Simple-homotopy theory, simple-homotopy type, parametrized complexity theory, simplicial complexes, (modified) dunce hat} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their diagram distances, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of persistence diagrams into Reproducing Kernel Hilbert Spaces, in which all computations can be performed easily. Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the metric properties of the feature map, i.e., the relationship between the Hilbert distance and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding. In this article, we study the possibility of embedding persistence diagrams into separable Hilbert spaces with bi-Lipschitz maps. In particular, we show that for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature, any lower bound must depend on the cardinalities of the persistence diagrams, and that when the Hilbert space is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.

Mathieu Carrière and Ulrich Bauer. On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carriere_et_al:LIPIcs.SoCG.2019.21, author = {Carri\`{e}re, Mathieu and Bauer, Ulrich}, title = {{On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.21}, URN = {urn:nbn:de:0030-drops-104259}, doi = {10.4230/LIPIcs.SoCG.2019.21}, annote = {Keywords: Topological Data Analysis, Persistence Diagrams, Hilbert space embedding} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov-Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.

Ulrich Bauer, Elizabeth Munch, and Yusu Wang. Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 461-475, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bauer_et_al:LIPIcs.SOCG.2015.461, author = {Bauer, Ulrich and Munch, Elizabeth and Wang, Yusu}, title = {{Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {461--475}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.461}, URN = {urn:nbn:de:0030-drops-51467}, doi = {10.4230/LIPIcs.SOCG.2015.461}, annote = {Keywords: Reeb graph, interleaving distance, functional distortion distance} }

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