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DOI: 10.4230/LIPIcs.SoCG.2026.3

Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs

Authors: Henry Adams, Sushovan Majhi, Fedor Manin, Žiga Virk, and Nicolò Zava

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

Let G be a finite, connected metric graph and let X be a subset of G. If X is sufficiently dense in G, we show that the Gromov-Hausdorff distance matches the Hausdorff distance, namely d_GH(G,X) = d_H(G,X). When the metric graph is the circle G = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/3 is the best possible. We lower bound the Gromov-Hausdorff distance d_GH(G,X) by the Hausdorff distance d_H(G,X) via a simple topological obstruction: the existence of a possibly discontinuous function f: G → X with too small distortion contradicts the connectedness of G.

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Henry Adams, Sushovan Majhi, Fedor Manin, Žiga Virk, and Nicolò Zava. Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{adams_et_al:LIPIcs.SoCG.2026.3,
  author =	{Adams, Henry and Majhi, Sushovan and Manin, Fedor and Virk, \v{Z}iga and Zava, Nicol\`{o}},
  title =	{{Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.3},
  URN =		{urn:nbn:de:0030-drops-258099},
  doi =		{10.4230/LIPIcs.SoCG.2026.3},
  annote =	{Keywords: Gromov-Hausdorff distance, distortion, connectedness, Borsuk-Ulam theorem}
}

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DOI: 10.4230/LIPIcs.SoCG.2024.73

Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

Authors: Sushovan Majhi

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

Abstract

For a closed Riemannian manifold ℳ and a metric space S with a small Gromov-Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale β > 0 at which the Vietoris-Rips complex of S is homotopy equivalent to ℳ. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale β in order to provide sampling conditions for S to be homotopy equivalent to ℳ. In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of β using the sectional curvatures and convexity radius of ℳ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris-Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris-Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.

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Sushovan Majhi. Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{majhi:LIPIcs.SoCG.2024.73,
  author =	{Majhi, Sushovan},
  title =	{{Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{73:1--73:16},
  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.2024.73},
  URN =		{urn:nbn:de:0030-drops-200188},
  doi =		{10.4230/LIPIcs.SoCG.2024.73},
  annote =	{Keywords: Vietoris-Rips complex, submanifold reconstruction, manifold reconstruction, Latschev’s theorem, homotopy Equivalence}
}

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Documents authored by Majhi, Sushovan

Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs

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Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

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