3 Search Results for "Sheth, Siddharth"


Document
Media Exposition
Greedy Permutations and Finite Voronoi Diagrams (Media Exposition)

Authors: Oliver A. Chubet, Paul Macnichol, Parth Parikh, Donald R. Sheehy, and Siddharth S. Sheth

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


Abstract
We illustrate the computation of a greedy permutation using finite Voronoi diagrams. We describe the neighbor graph, which is a sparse graph data structure that facilitates efficient point location to insert a new Voronoi cell. This data structure is not dependent on a Euclidean metric space. The greedy permutation is computed in O(nlog Δ) time for low-dimensional data using this method [Sariel Har-Peled and Manor Mendel, 2006; Donald R. Sheehy, 2020].

Cite as

Oliver A. Chubet, Paul Macnichol, Parth Parikh, Donald R. Sheehy, and Siddharth S. Sheth. Greedy Permutations and Finite Voronoi Diagrams (Media Exposition). In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 64:1-64:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{chubet_et_al:LIPIcs.SoCG.2023.64,
  author =	{Chubet, Oliver A. and Macnichol, Paul and Parikh, Parth and Sheehy, Donald R. and Sheth, Siddharth S.},
  title =	{{Greedy Permutations and Finite Voronoi Diagrams}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{64:1--64:5},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.64},
  URN =		{urn:nbn:de:0030-drops-179146},
  doi =		{10.4230/LIPIcs.SoCG.2023.64},
  annote =	{Keywords: greedy permutation, Voronoi diagrams}
}
Document
Nearly-Doubling Spaces of Persistence Diagrams

Authors: Donald R. Sheehy and Siddharth S. Sheth

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


Abstract
The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space. We introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded k-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example). We also prove bounds on the dimension of the k-point bottleneck space over such metrics. The notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams.

Cite as

Donald R. Sheehy and Siddharth S. Sheth. Nearly-Doubling Spaces of Persistence Diagrams. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{sheehy_et_al:LIPIcs.SoCG.2022.60,
  author =	{Sheehy, Donald R. and Sheth, Siddharth S.},
  title =	{{Nearly-Doubling Spaces of Persistence Diagrams}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{60:1--60:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.60},
  URN =		{urn:nbn:de:0030-drops-160686},
  doi =		{10.4230/LIPIcs.SoCG.2022.60},
  annote =	{Keywords: Topological Data Analysis, Persistence Diagrams, Gromov-Hausdorff Distance}
}
Document
Sketching Persistence Diagrams

Authors: Donald R. Sheehy and Siddharth Sheth

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)


Abstract
Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams - a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.

Cite as

Donald R. Sheehy and Siddharth Sheth. Sketching Persistence Diagrams. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Copy BibTex To Clipboard

@InProceedings{sheehy_et_al:LIPIcs.SoCG.2021.57,
  author =	{Sheehy, Donald R. and Sheth, Siddharth},
  title =	{{Sketching Persistence Diagrams}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.57},
  URN =		{urn:nbn:de:0030-drops-138569},
  doi =		{10.4230/LIPIcs.SoCG.2021.57},
  annote =	{Keywords: Bottleneck Distance, Persistent Homology, Approximate Persistence Diagrams}
}
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