7 Search Results for "Lazarus, Francis"


Document
Making Multicurves Cross Minimally on Surfaces

Authors: Loïc Dubois

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
On an orientable surface S, consider a collection Γ of closed curves. The (geometric) intersection number i_S(Γ) is the minimum number of self-intersections that a collection Γ' can have, where Γ' results from a continuous deformation (homotopy) of Γ. We provide algorithms that compute i_S(Γ) and such a Γ', assuming that Γ is given by a collection of closed walks of length n in a graph M cellularly embedded on S, in O(n log n) time when M and S are fixed. The state of the art is a paper of Despré and Lazarus [SoCG 2017, J. ACM 2019], who compute i_S(Γ) in O(n²) time, and Γ' in O(n⁴) time if Γ is a single closed curve. Our result is more general since we can put an arbitrary number of closed curves in minimal position. Also, our algorithms are quasi-linear in n instead of quadratic and quartic. Most importantly, our proofs are simpler, shorter, and more structured. We use techniques from two-dimensional topology and from the theory of hyperbolic surfaces. Most notably, we prove a new property of the reducing triangulations introduced by Colin de Verdière, Despré, and Dubois [SODA 2024], reducing our problem to the case of surfaces with boundary. As a key subroutine, we rely on an algorithm of Fulek and Tóth [JCO 2020].

Cite as

Loïc Dubois. Making Multicurves Cross Minimally on Surfaces. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{dubois:LIPIcs.ESA.2024.50,
  author =	{Dubois, Lo\"{i}c},
  title =	{{Making Multicurves Cross Minimally on Surfaces}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{50:1--50:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.50},
  URN =		{urn:nbn:de:0030-drops-211216},
  doi =		{10.4230/LIPIcs.ESA.2024.50},
  annote =	{Keywords: Algorithms, Topology, Surfaces, Closed Curves, Geometric Intersection Number}
}
Document
Algorithms for Length Spectra of Combinatorial Tori

Authors: Vincent Delecroix, Matthijs Ebbens, Francis Lazarus, and Ivan Yakovlev

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


Abstract
Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order. In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity n in time O(n² log log n) so that, given a cycle with 𝓁 edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in O(𝓁+log n) time. Moreover, given any positive integer k, the first k values of its unmarked length spectrum can be computed in time O(k log n). Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.

Cite as

Vincent Delecroix, Matthijs Ebbens, Francis Lazarus, and Ivan Yakovlev. Algorithms for Length Spectra of Combinatorial Tori. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{delecroix_et_al:LIPIcs.SoCG.2023.26,
  author =	{Delecroix, Vincent and Ebbens, Matthijs and Lazarus, Francis and Yakovlev, Ivan},
  title =	{{Algorithms for Length Spectra of Combinatorial Tori}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{26:1--26:16},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.26},
  URN =		{urn:nbn:de:0030-drops-178765},
  doi =		{10.4230/LIPIcs.SoCG.2023.26},
  annote =	{Keywords: graphs on surfaces, length spectrum, polyhedral norm}
}
Document
Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

Authors: Jean Chartier and Arnaud de Mesmay

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


Abstract
A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most π on both sides. While the existence of a simple closed quasigeodesic on a convex polyhedron has been proved by Pogorelov in 1949, finding a polynomial-time algorithm to compute such a simple closed quasigeodesic has been repeatedly posed as an open problem. Our first contribution is to propose an extended definition of quasigeodesics in the intrinsic setting of (not necessarily convex) polyhedral spheres, and to prove the existence of a weakly simple closed quasigeodesic in such a setting. Our proof does not proceed via an approximation by smooth surfaces, but relies on an adapation of the disk flow of Hass and Scott to the context of polyhedral surfaces. Our second result is to leverage this existence theorem to provide a finite algorithm to compute a weakly simple closed quasigeodesic on a polyhedral sphere. On a convex polyhedron, our algorithm computes a simple closed quasigeodesic, solving an open problem of Demaine, Hersterberg and Ku.

Cite as

Jean Chartier and Arnaud de Mesmay. Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chartier_et_al:LIPIcs.SoCG.2022.27,
  author =	{Chartier, Jean and de Mesmay, Arnaud},
  title =	{{Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{27:1--27:16},
  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.2022.27},
  URN =		{urn:nbn:de:0030-drops-160350},
  doi =		{10.4230/LIPIcs.SoCG.2022.27},
  annote =	{Keywords: Quasigeodesic, polyhedron, curve-shortening process, disk flow, weakly simple}
}
Document
Short Topological Decompositions of Non-Orientable Surfaces

Authors: Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay

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


Abstract
We investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. This result confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.

Cite as

Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay. Short Topological Decompositions of Non-Orientable Surfaces. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{fuladi_et_al:LIPIcs.SoCG.2022.41,
  author =	{Fuladi, Niloufar and Hubard, Alfredo and de Mesmay, Arnaud},
  title =	{{Short Topological Decompositions of Non-Orientable Surfaces}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{41:1--41:16},
  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.2022.41},
  URN =		{urn:nbn:de:0030-drops-160492},
  doi =		{10.4230/LIPIcs.SoCG.2022.41},
  annote =	{Keywords: Computational topology, embedded graph, non-orientable surface, joint crossing number, canonical system of loop, surface decomposition}
}
Document
A Universal Triangulation for Flat Tori

Authors: Francis Lazarus and Florent Tallerie

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


Abstract
A result due to Burago and Zalgaller states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space 𝔼³. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially constructive, to produce PL isometric embeddings of flat tori. In practice, the resulting embeddings have a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller and on recent works by Arnoux et al., we exhibit a universal triangulation with 5974 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.

Cite as

Francis Lazarus and Florent Tallerie. A Universal Triangulation for Flat Tori. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lazarus_et_al:LIPIcs.SoCG.2022.53,
  author =	{Lazarus, Francis and Tallerie, Florent},
  title =	{{A Universal Triangulation for Flat Tori}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{53:1--53:18},
  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.2022.53},
  URN =		{urn:nbn:de:0030-drops-160617},
  doi =		{10.4230/LIPIcs.SoCG.2022.53},
  annote =	{Keywords: Triangulation, flat torus, isometric embedding}
}
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.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
Computing the Geometric Intersection Number of Curves

Authors: Vincent Despré and Francis Lazarus

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)


Abstract
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time.

Cite as

Vincent Despré and Francis Lazarus. Computing the Geometric Intersection Number of Curves. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{despre_et_al:LIPIcs.SoCG.2017.35,
  author =	{Despr\'{e}, Vincent and Lazarus, Francis},
  title =	{{Computing the Geometric Intersection Number of Curves}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{35:1--35:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.35},
  URN =		{urn:nbn:de:0030-drops-71838},
  doi =		{10.4230/LIPIcs.SoCG.2017.35},
  annote =	{Keywords: computational topology, curves on surfaces, combinatorial geodesic}
}
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