4 Search Results for "Hou, Tao"


Document
Computing Zigzag Vineyard Efficiently Including Expansions and Contractions

Authors: Tamal K. Dey and Tao Hou

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


Abstract
Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. [Cohen-Steiner et al., 2006]. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm [Tamal K. Dey and Tao Hou, 2022], expansions and contractions pose difficulty in breaking the barrier of cubic complexity [Dey and Hou, 2022]. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case.

Cite as

Tamal K. Dey and Tao Hou. Computing Zigzag Vineyard Efficiently Including Expansions and Contractions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{dey_et_al:LIPIcs.SoCG.2024.49,
  author =	{Dey, Tamal K. and Hou, Tao},
  title =	{{Computing Zigzag Vineyard Efficiently Including Expansions and Contractions}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{49:1--49:15},
  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.2024.49},
  URN =		{urn:nbn:de:0030-drops-199942},
  doi =		{10.4230/LIPIcs.SoCG.2024.49},
  annote =	{Keywords: zigzag persistence, vines and vineyard, update operations}
}
Document
Survey
Semantic Web: Past, Present, and Future

Authors: Ansgar Scherp, Gerd Groener, Petr Škoda, Katja Hose, and Maria-Esther Vidal

Published in: TGDK, Volume 2, Issue 1 (2024): Special Issue on Trends in Graph Data and Knowledge - Part 2. Transactions on Graph Data and Knowledge, Volume 2, Issue 1


Abstract
Ever since the vision was formulated, the Semantic Web has inspired many generations of innovations. Semantic technologies have been used to share vast amounts of information on the Web, enhance them with semantics to give them meaning, and enable inference and reasoning on them. Throughout the years, semantic technologies, and in particular knowledge graphs, have been used in search engines, data integration, enterprise settings, and machine learning. In this paper, we recap the classical concepts and foundations of the Semantic Web as well as modern and recent concepts and applications, building upon these foundations. The classical topics we cover include knowledge representation, creating and validating knowledge on the Web, reasoning and linking, and distributed querying. We enhance this classical view of the so-called "Semantic Web Layer Cake" with an update of recent concepts that include provenance, security and trust, as well as a discussion of practical impacts from industry-led contributions. We conclude with an outlook on the future directions of the Semantic Web. This is a living document. If you like to contribute, please contact the first author and visit: https://github.com/ascherp/semantic-web-primer

Cite as

Ansgar Scherp, Gerd Groener, Petr Škoda, Katja Hose, and Maria-Esther Vidal. Semantic Web: Past, Present, and Future. In Special Issue on Trends in Graph Data and Knowledge - Part 2. Transactions on Graph Data and Knowledge (TGDK), Volume 2, Issue 1, pp. 3:1-3:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@Article{scherp_et_al:TGDK.2.1.3,
  author =	{Scherp, Ansgar and Groener, Gerd and \v{S}koda, Petr and Hose, Katja and Vidal, Maria-Esther},
  title =	{{Semantic Web: Past, Present, and Future}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{3:1--3:37},
  ISSN =	{2942-7517},
  year =	{2024},
  volume =	{2},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.2.1.3},
  URN =		{urn:nbn:de:0030-drops-198607},
  doi =		{10.4230/TGDK.2.1.3},
  annote =	{Keywords: Linked Open Data, Semantic Web Graphs, Knowledge Graphs}
}
Document
Fast Computation of Zigzag Persistence

Authors: Tamal K. Dey and Tao Hou

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a Δ-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.

Cite as

Tamal K. Dey and Tao Hou. Fast Computation of Zigzag Persistence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{dey_et_al:LIPIcs.ESA.2022.43,
  author =	{Dey, Tamal K. and Hou, Tao},
  title =	{{Fast Computation of Zigzag Persistence}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.43},
  URN =		{urn:nbn:de:0030-drops-169813},
  doi =		{10.4230/LIPIcs.ESA.2022.43},
  annote =	{Keywords: zigzag persistence, persistent homology, fast computation}
}
Document
Computing Zigzag Persistence on Graphs in Near-Linear Time

Authors: Tamal K. Dey and Tao Hou

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


Abstract
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general O(m^ω) time complexity are not known, where ω < 2.37286 is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with m additions and deletions on a graph with n vertices and edges, the algorithm for 0-dimension runs in O(mlog² n+mlog m) time and the algorithm for 1-dimension runs in O(mlog⁴ n) time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a 1-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for 0-dimension to compute the (p-1)-dimensional zigzag persistence for ℝ^p-embedded complexes in O(mlog² n+mlog m+nlog n) time.

Cite as

Tamal K. Dey and Tao Hou. Computing Zigzag Persistence on Graphs in Near-Linear Time. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Copy BibTex To Clipboard

@InProceedings{dey_et_al:LIPIcs.SoCG.2021.30,
  author =	{Dey, Tamal K. and Hou, Tao},
  title =	{{Computing Zigzag Persistence on Graphs in Near-Linear Time}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{30:1--30:15},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.30},
  URN =		{urn:nbn:de:0030-drops-138292},
  doi =		{10.4230/LIPIcs.SoCG.2021.30},
  annote =	{Keywords: persistent homology, zigzag persistence, graph filtration, dynamic networks}
}
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