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DOI: 10.4230/LIPIcs.SoCG.2022.14

Quasi-Universality of Reeb Graph Distances

Authors: Ulrich Bauer, Håvard Bakke Bjerkevik, and Benedikt Fluhr

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

Abstract

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.

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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}
}

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DOI: 10.4230/LIPIcs.SoCG.2018.13

Computational Complexity of the Interleaving Distance

Authors: Håvard Bakke Bjerkevik and Magnus Bakke Botnan

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.

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Håvard Bakke Bjerkevik and Magnus Bakke Botnan. Computational Complexity of the Interleaving Distance. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bjerkevik_et_al:LIPIcs.SoCG.2018.13,
  author =	{Bjerkevik, H\r{a}vard Bakke and Botnan, Magnus Bakke},
  title =	{{Computational Complexity of the Interleaving Distance}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.13},
  URN =		{urn:nbn:de:0030-drops-87268},
  doi =		{10.4230/LIPIcs.SoCG.2018.13},
  annote =	{Keywords: Persistent Homology, Interleavings, NP-hard}
}

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Documents authored by Bjerkevik, Håvard Bakke

Quasi-Universality of Reeb Graph Distances

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Computational Complexity of the Interleaving Distance

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