7 Search Results for "Chambers, Erin Wolf"


Document
On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

Authors: Erin Wolf Chambers, Salman Parsa, and Hannah Schreiber

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


Abstract
Homology features of spaces which appear in applications, for instance 3D meshes, are among the most important topological properties of these objects. Given a non-trivial cycle in a homology class, we consider the problem of computing a representative in that homology class which is optimal. We study two measures of optimality, namely, the lexicographic order of cycles (the lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We give a simple algorithm for computing the lex-optimal cycle for a 1-homology class in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-manifolds of size n² in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an n × n sparse matrix. From this reduction, we deduce several hardness results. Most notably, we show that for 3-manifolds given as a subset of the 3-space of size n², persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in O(n² log n) time. This is the first such distinction between these two computations. Moreover, it follows that the same disparity exists between the height persistent homology computation and general sub-level set persistent homology computation for simplicial complexes in the 3-space.

Cite as

Erin Wolf Chambers, Salman Parsa, and Hannah Schreiber. On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chambers_et_al:LIPIcs.SoCG.2022.25,
  author =	{Chambers, Erin Wolf and Parsa, Salman and Schreiber, Hannah},
  title =	{{On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{25:1--25:15},
  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.2022.25},
  URN =		{urn:nbn:de:0030-drops-160338},
  doi =		{10.4230/LIPIcs.SoCG.2022.25},
  annote =	{Keywords: computational topology, bottleneck optimal cycles, homology}
}
Document
A Family of Metrics from the Truncated Smoothing of Reeb Graphs

Authors: Erin Wolf Chambers, Elizabeth Munch, and Tim Ophelders

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


Abstract
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter τ. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ≤ τ ≤ 2ε, where ε is the smoothing parameter. Then, for the restriction of τ ∈ [0,ε], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ∈ [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ∈ [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m' ∈ [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.

Cite as

Erin Wolf Chambers, Elizabeth Munch, and Tim Ophelders. A Family of Metrics from the Truncated Smoothing of Reeb Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{chambers_et_al:LIPIcs.SoCG.2021.22,
  author =	{Chambers, Erin Wolf and Munch, Elizabeth and Ophelders, Tim},
  title =	{{A Family of Metrics from the Truncated Smoothing of Reeb Graphs}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{22:1--22:17},
  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.2021.22},
  URN =		{urn:nbn:de:0030-drops-138218},
  doi =		{10.4230/LIPIcs.SoCG.2021.22},
  annote =	{Keywords: Reeb graphs, interleaving distance, graphical signatures}
}
Document
Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

Authors: Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, and Salman Parsa

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


Abstract
In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve γ, and a collection of disjoint normal curves Δ, there is a polynomial-time algorithm to decide if γ lies in the normal subgroup generated by components of Δ in the fundamental group of the surface after attaching the curves to a basepoint.

Cite as

Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, and Salman Parsa. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{chambers_et_al:LIPIcs.SoCG.2021.23,
  author =	{Chambers, Erin Wolf and Lazarus, Francis and de Mesmay, Arnaud and Parsa, Salman},
  title =	{{Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{23:1--23:16},
  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.2021.23},
  URN =		{urn:nbn:de:0030-drops-138223},
  doi =		{10.4230/LIPIcs.SoCG.2021.23},
  annote =	{Keywords: 3-manifolds, surfaces, low-dimensional topology, contractibility, compressed curves}
}
Document
Low-Stretch Spanning Trees of Graphs with Bounded Width

Authors: Glencora Borradaile, Erin Wolf Chambers, David Eppstein, William Maxwell, and Amir Nayyeri

Published in: LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)


Abstract
We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph G with a linear arrangement of bandwidth b can be embedded into a distribution T of spanning trees such that the expected stretch of each edge of G is O(b²). Our proof implies a linear time algorithm for sampling from T. Therefore, we have a linear time algorithm that finds a spanning tree of G with average stretch O(b²) with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of G with average stretch O(b³). For graphs of cutwidth c, we construct a spanning tree with stretch O(c²) in linear time. Finally, when G has treewidth k we provide a dynamic programming algorithm computing a minimum stretch spanning tree of G that runs in polynomial time with respect to the number of vertices of G.

Cite as

Glencora Borradaile, Erin Wolf Chambers, David Eppstein, William Maxwell, and Amir Nayyeri. Low-Stretch Spanning Trees of Graphs with Bounded Width. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{borradaile_et_al:LIPIcs.SWAT.2020.15,
  author =	{Borradaile, Glencora and Chambers, Erin Wolf and Eppstein, David and Maxwell, William and Nayyeri, Amir},
  title =	{{Low-Stretch Spanning Trees of Graphs with Bounded Width}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Albers, Susanne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.15},
  URN =		{urn:nbn:de:0030-drops-122622},
  doi =		{10.4230/LIPIcs.SWAT.2020.15},
  annote =	{Keywords: Treewidth, low-stretch spanning tree, fundamental cycle basis}
}
Document
Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352)

Authors: Maarten Löffler, Anna Lubiw, Saul Schleimer, and Erin Moriarty Wolf Chambers

Published in: Dagstuhl Reports, Volume 9, Issue 8 (2020)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 19352 ``Computation in Low-Dimensional Geometry and Topology''. The seminar participants investigated problems in: knot theory, trajectory analysis, algorithmic topology, computational geometry of curves, and graph drawing, with an emphasis on how low-dimensional structures change over time.

Cite as

Maarten Löffler, Anna Lubiw, Saul Schleimer, and Erin Moriarty Wolf Chambers. Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352). In Dagstuhl Reports, Volume 9, Issue 8, pp. 84-112, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@Article{loffler_et_al:DagRep.9.8.84,
  author =	{L\"{o}ffler, Maarten and Lubiw, Anna and Schleimer, Saul and Wolf Chambers, Erin Moriarty},
  title =	{{Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352)}},
  pages =	{84--112},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{9},
  number =	{8},
  editor =	{L\"{o}ffler, Maarten and Lubiw, Anna and Schleimer, Saul and Wolf Chambers, Erin Moriarty},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.9.8.84},
  URN =		{urn:nbn:de:0030-drops-117734},
  doi =		{10.4230/DagRep.9.8.84},
  annote =	{Keywords: Geometric topology, Graph Drawing, Computational Geometry}
}
Document
Applications of Topology to the Analysis of 1-Dimensional Objects (Dagstuhl Seminar 17072)

Authors: Benjamin Burton, Maarten Löffler, Carola Wenk, and Erin Moriarty Wolf Chambers

Published in: Dagstuhl Reports, Volume 7, Issue 2 (2017)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 17072 "Applications of Topology to the Analysis of 1-Dimensional Objects".

Cite as

Benjamin Burton, Maarten Löffler, Carola Wenk, and Erin Moriarty Wolf Chambers. Applications of Topology to the Analysis of 1-Dimensional Objects (Dagstuhl Seminar 17072). In Dagstuhl Reports, Volume 7, Issue 2, pp. 64-88, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@Article{burton_et_al:DagRep.7.2.64,
  author =	{Burton, Benjamin and L\"{o}ffler, Maarten and Wenk, Carola and Wolf Chambers, Erin Moriarty},
  title =	{{Applications of Topology to the Analysis of 1-Dimensional Objects (Dagstuhl Seminar 17072)}},
  pages =	{64--88},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{7},
  number =	{2},
  editor =	{Burton, Benjamin and L\"{o}ffler, Maarten and Wenk, Carola and Wolf Chambers, Erin Moriarty},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.7.2.64},
  URN =		{urn:nbn:de:0030-drops-73536},
  doi =		{10.4230/DagRep.7.2.64},
  annote =	{Keywords: curves, graph drawing, homotopy, knot theory}
}
Document
Minimum Cycle and Homology Bases of Surface Embedded Graphs

Authors: Glencora Borradaile, Erin Wolf Chambers, Kyle Fox, and Amir Nayyeri

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)


Abstract
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 1-dimensional (Z_2)-homology classes) of an undirected graph embedded on an orientable surface of genus g. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic O(n^omega + 2^2g n^2)-time algorithm. The best known existing algorithms for surface embedded graphs are those for general sparse graphs: an O(n^omega) time Monte Carlo algorithm [Amaldi et. al., ESA'09] and a deterministic O(n^3) time algorithm [Mehlhorn and Michail, TALG'09]. For the minimum homology basis problem, we give an O(g^3 n log n)-time algorithm, improving on existing algorithms for many values of g and n.

Cite as

Glencora Borradaile, Erin Wolf Chambers, Kyle Fox, and Amir Nayyeri. Minimum Cycle and Homology Bases of Surface Embedded Graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{borradaile_et_al:LIPIcs.SoCG.2016.23,
  author =	{Borradaile, Glencora and Chambers, Erin Wolf and Fox, Kyle and Nayyeri, Amir},
  title =	{{Minimum Cycle and Homology Bases of Surface Embedded Graphs}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.23},
  URN =		{urn:nbn:de:0030-drops-59152},
  doi =		{10.4230/LIPIcs.SoCG.2016.23},
  annote =	{Keywords: Cycle basis, Homology basis, Topological graph theory}
}
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