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Documents authored by Cabello, Sergio

Document
DOI: 10.4230/LIPIcs.ESA.2023.28
On k-Means for Segments and Polylines

Authors: Sergio Cabello and Panos Giannopoulos

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
We study the problem of k-means clustering in the space of straight-line segments in ℝ² under the Hausdorff distance. For this problem, we give a (1+ε)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n + ε^{-O(k)} + ε^{-O(k)} ⋅ log^O(k) (ε^{-1})). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron [Antoine Vigneron, 2014] to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg [Dan Feldman and Michael Langberg, 2011; Feldman et al., 2020]. Our results can be extended to polylines of constant complexity with a running time of O(n + ε^{-O(k)}).
Cite as

Sergio Cabello and Panos Giannopoulos. On k-Means for Segments and Polylines. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cabello_et_al:LIPIcs.ESA.2023.28,
  author =	{Cabello, Sergio and Giannopoulos, Panos},
  title =	{{On k-Means for Segments and Polylines}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{28:1--28:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.28},
  URN =		{urn:nbn:de:0030-drops-186812},
  doi =		{10.4230/LIPIcs.ESA.2023.28},
  annote =	{Keywords: k-means clustering, segments, polylines, Hausdorff distance, Fr\'{e}chet mean}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.23
Long Plane Trees

Authors: Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec

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

Abstract
In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).
Cite as

Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23,
  author =	{Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef},
  title =	{{Long Plane Trees}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{23:1--23:17},
  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.2022.23},
  URN =		{urn:nbn:de:0030-drops-160311},
  doi =		{10.4230/LIPIcs.SoCG.2022.23},
  annote =	{Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2020.2
Some Open Problems in Computational Geometry (Invited Talk)

Authors: Sergio Cabello

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
In this paper we shall encounter three open problems in Computational Geometry that are, in my opinion, interesting for a general audience interested in algorithms.
Cite as

Sergio Cabello. Some Open Problems in Computational Geometry (Invited Talk). In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 2:1-2:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cabello:LIPIcs.MFCS.2020.2,
  author =	{Cabello, Sergio},
  title =	{{Some Open Problems in Computational Geometry}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{2:1--2:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.2},
  URN =		{urn:nbn:de:0030-drops-126734},
  doi =		{10.4230/LIPIcs.MFCS.2020.2},
  annote =	{Keywords: barrier resilience, maximum matching, geometric graphs, fixed-parameter tractability, stochastic computational geometry}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.SoCG.2020
LIPIcs, Volume 164, SoCG 2020, Complete Volume

Authors: Sergio Cabello and Danny Z. Chen

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
LIPIcs, Volume 164, SoCG 2020, Complete Volume
Cite as

36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1-1222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{cabello_et_al:LIPIcs.SoCG.2020,
  title =	{{LIPIcs, Volume 164, SoCG 2020, Complete Volume}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{1--1222},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020},
  URN =		{urn:nbn:de:0030-drops-121576},
  doi =		{10.4230/LIPIcs.SoCG.2020},
  annote =	{Keywords: LIPIcs, Volume 164, SoCG 2020, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.SoCG.2020.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Sergio Cabello and Danny Z. Chen

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2020.0,
  author =	{Cabello, Sergio and Chen, Danny Z.},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{0:i--0:xx},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.0},
  URN =		{urn:nbn:de:0030-drops-121587},
  doi =		{10.4230/LIPIcs.SoCG.2020.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.31
Maximum Matchings in Geometric Intersection Graphs

Authors: Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.
Cite as

Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonnet_et_al:LIPIcs.STACS.2020.31,
  author =	{Bonnet, \'{E}douard and Cabello, Sergio and Mulzer, Wolfgang},
  title =	{{Maximum Matchings in Geometric Intersection Graphs}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.31},
  URN =		{urn:nbn:de:0030-drops-118926},
  doi =		{10.4230/LIPIcs.STACS.2020.31},
  annote =	{Keywords: computational geometry, geometric intersection graph, maximum matching, disk graph, unit-disk graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.20
Computing Shapley Values in the Plane

Authors: Sergio Cabello and Timothy M. Chan

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.
Cite as

Sergio Cabello and Timothy M. Chan. Computing Shapley Values in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2019.20,
  author =	{Cabello, Sergio and Chan, Timothy M.},
  title =	{{Computing Shapley Values in the Plane}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.20},
  URN =		{urn:nbn:de:0030-drops-104244},
  doi =		{10.4230/LIPIcs.SoCG.2019.20},
  annote =	{Keywords: Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.6
The Reverse Kakeya Problem

Authors: Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, and Sang Duk Yoon

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.
Cite as

Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, and Sang Duk Yoon. The Reverse Kakeya Problem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bae_et_al:LIPIcs.SoCG.2018.6,
  author =	{Bae, Sang Won and Cabello, Sergio and Cheong, Otfried and Choi, Yoonsung and Stehn, Fabian and Yoon, Sang Duk},
  title =	{{The Reverse Kakeya Problem}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{6:1--6:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.6},
  URN =		{urn:nbn:de:0030-drops-87199},
  doi =		{10.4230/LIPIcs.SoCG.2018.6},
  annote =	{Keywords: Kakeya problem, convex, isodynamic point, turning}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.22
Maximum Volume Subset Selection for Anchored Boxes

Authors: Karl Bringmann, Sergio Cabello, and Michael T. M. Emmerich

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

Abstract
Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that: * The problem is NP-hard already in 3 dimensions. * In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm. * For any constant dimension d, we give an efficient polynomial-time approximation scheme.
Cite as

Karl Bringmann, Sergio Cabello, and Michael T. M. Emmerich. Maximum Volume Subset Selection for Anchored Boxes. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2017.22,
  author =	{Bringmann, Karl and Cabello, Sergio and Emmerich, Michael T. M.},
  title =	{{Maximum Volume Subset Selection for Anchored Boxes}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{22:1--22:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.22},
  URN =		{urn:nbn:de:0030-drops-72011},
  doi =		{10.4230/LIPIcs.SoCG.2017.22},
  annote =	{Keywords: geometric optimization, subset selection, hypervolume indicator, Klee’s 23 measure problem, boxes, NP-hardness, PTAS}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.18
The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

Authors: Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).
Cite as

Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{burton_et_al:LIPIcs.STACS.2017.18,
  author =	{Burton, Benjamin and Cabello, Sergio and Kratsch, Stefan and Pettersson, William},
  title =	{{The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.18},
  URN =		{urn:nbn:de:0030-drops-70156},
  doi =		{10.4230/LIPIcs.STACS.2017.18},
  annote =	{Keywords: computational topology, parameterized complexity, simplicial complex}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.240
Finding All Maximal Subsequences with Hereditary Properties

Authors: Drago Bokal, Sergio Cabello, and David Eppstein

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results: - In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1. - In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1. - In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction. The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.
Cite as

Drago Bokal, Sergio Cabello, and David Eppstein. Finding All Maximal Subsequences with Hereditary Properties. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 240-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bokal_et_al:LIPIcs.SOCG.2015.240,
  author =	{Bokal, Drago and Cabello, Sergio and Eppstein, David},
  title =	{{Finding All Maximal Subsequences with Hereditary Properties}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{240--254},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.240},
  URN =		{urn:nbn:de:0030-drops-51132},
  doi =		{10.4230/LIPIcs.SOCG.2015.240},
  annote =	{Keywords: convex hull, diameter, monotone path, sequence of points, trajectory}
}
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