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

Let P be a simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. Given a set S of m sites being a subset of the vertices of P, we present the first randomized algorithm to compute the geodesic farthest-point Voronoi diagram of S in P running in expected O(n + m) time. That is, a partition of P into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. This algorithm can be extended to run in expected O(n + m log m) time when S is an arbitrary set of m sites contained in P.

Luis Barba. Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barba:LIPIcs.SoCG.2019.12, author = {Barba, Luis}, title = {{Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {12:1--12:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.12}, URN = {urn:nbn:de:0030-drops-104161}, doi = {10.4230/LIPIcs.SoCG.2019.12}, annote = {Keywords: Geodesic distance, simple polygons, farthest-point Voronoi diagram} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

We consider asymmetric convex intersection testing (ACIT).
Let P subset R^d be a set of n points and H a set of n halfspaces in d dimensions. We denote by {ch(P)} the polytope obtained by taking the convex hull of P, and by {fh(H)} the polytope obtained by taking the intersection of the halfspaces in H. Our goal is to decide whether the intersection of H and the convex hull of P are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before.
We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on n and m, whose running time depends reasonably on the dimension d.

Luis Barba and Wolfgang Mulzer. Asymmetric Convex Intersection Testing. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barba_et_al:OASIcs.SOSA.2019.9, author = {Barba, Luis and Mulzer, Wolfgang}, title = {{Asymmetric Convex Intersection Testing}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {9:1--9:14}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.9}, URN = {urn:nbn:de:0030-drops-100358}, doi = {10.4230/OASIcs.SOSA.2019.9}, annote = {Keywords: polytope intersection, LP-type problem, randomized algorithm} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the "rubber band" conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a different, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x,y in X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite differently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that suffice to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n^3h^{3+epsilon}) time, for any constant epsilon > 0.

Luis Barba, Michael Hoffmann, Matias Korman, and Alexander Pilz. Convex Hulls in Polygonal Domains. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{barba_et_al:LIPIcs.SWAT.2018.8, author = {Barba, Luis and Hoffmann, Michael and Korman, Matias and Pilz, Alexander}, title = {{Convex Hulls in Polygonal Domains}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.8}, URN = {urn:nbn:de:0030-drops-88343}, doi = {10.4230/LIPIcs.SWAT.2018.8}, annote = {Keywords: geometric graph, polygonal domain, geodesic hull, shortest path} }

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

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11/6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry.
We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+e}) that solve 3POL, and that 3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on o((log n)^{1/6}/(log log n)^{1/2}) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.

Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon. Subquadratic Algorithms for Algebraic Generalizations of 3SUM. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barba_et_al:LIPIcs.SoCG.2017.13, author = {Barba, Luis and Cardinal, Jean and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien and Solomon, Noam}, title = {{Subquadratic Algorithms for Algebraic Generalizations of 3SUM}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {13:1--13:15}, 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.2017.13}, URN = {urn:nbn:de:0030-drops-72214}, doi = {10.4230/LIPIcs.SoCG.2017.13}, annote = {Keywords: 3SUM, subquadratic algorithms, general position testing, range searching, dominance reporting, polynomial curves} }

Document

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

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull.
We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result.
The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Sarah R. Allen, Luis Barba, John Iacono, and Stefan Langerman. Incremental Voronoi diagrams. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{allen_et_al:LIPIcs.SoCG.2016.15, author = {Allen, Sarah R. and Barba, Luis and Iacono, John and Langerman, Stefan}, title = {{Incremental Voronoi diagrams}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {15:1--15:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.15}, URN = {urn:nbn:de:0030-drops-59079}, doi = {10.4230/LIPIcs.SoCG.2016.15}, annote = {Keywords: Voronoi diagrams, dynamic data structures, Delaunay triangulation} }

Document

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

Given a set of sites (points) in a simple polygon, the farthest-point geodesic Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an O((n+m)loglogn)-time algorithm to compute the farthest-point geodesic Voronoi diagram for m sites lying on the boundary of a simple n-gon.

Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The Farthest-Point Geodesic Voronoi Diagram of Points on the Boundary of a Simple Polygon. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2016.56, author = {Oh, Eunjin and Barba, Luis and Ahn, Hee-Kap}, title = {{The Farthest-Point Geodesic Voronoi Diagram of Points on the Boundary of a Simple Polygon}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {56:1--56:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.56}, URN = {urn:nbn:de:0030-drops-59481}, doi = {10.4230/LIPIcs.SoCG.2016.56}, annote = {Keywords: Geodesic distance, simple polygons, farthest-point Voronoi diagram} }

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

Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

Hee Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 209-223, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{ahn_et_al:LIPIcs.SOCG.2015.209, author = {Ahn, Hee Kap and Barba, Luis and Bose, Prosenjit and De Carufel, Jean-Lou and Korman, Matias and Oh, Eunjin}, title = {{A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {209--223}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.209}, URN = {urn:nbn:de:0030-drops-51448}, doi = {10.4230/LIPIcs.SOCG.2015.209}, annote = {Keywords: Geodesic distance, facility location, 1-center problem, simple polygons} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memory-constrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Theta(n), we can modify it so that it runs in O(n^2/2^s) time using a workspace of O(s) variables (for any s \in o(log n)) or O(n log n/log p)$ time using O(p log n/log p) variables (for any 2 <= p <= n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.

Luis Barba, Matias Korman, Stefan Langerman, Rodrigo I. Silveira, and Kunihiko Sadakane. Space-Time Trade-offs for Stack-Based Algorithms. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 281-292, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{barba_et_al:LIPIcs.STACS.2013.281, author = {Barba, Luis and Korman, Matias and Langerman, Stefan and Silveira, Rodrigo I. and Sadakane, Kunihiko}, title = {{Space-Time Trade-offs for Stack-Based Algorithms}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {281--292}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.281}, URN = {urn:nbn:de:0030-drops-39411}, doi = {10.4230/LIPIcs.STACS.2013.281}, annote = {Keywords: space-time trade-off, constant workspace, stack algorithms} }

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