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Media Exposition

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

There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric objects based on polygonal geometries. These include Macbeath regions, metric balls in the forward and reverse Funk distance, metric balls in the Hilbert metric, polar bodies, the minimum enclosing ball of a point set, and minimum spanning trees in both the Funk and Hilbert metrics. We also include a number of utilities on convex polygons, including union, intersection, subtraction, and Minkowski sum (previously implemented as a CGAL Ipelet).

Nithin Parepally, Ainesh Chatterjee, Auguste H. Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, and David M. Mount. Ipelets for the Convex Polygonal Geometry (Media Exposition). In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 92:1-92:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{parepally_et_al:LIPIcs.SoCG.2024.92, author = {Parepally, Nithin and Chatterjee, Ainesh and Gezalyan, Auguste H. and Du, Hongyang and Mangla, Sukrit and Wu, Kenny and Hwang, Sarah and Mount, David M.}, title = {{Ipelets for the Convex Polygonal Geometry}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {92:1--92:7}, 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.2024.92}, URN = {urn:nbn:de:0030-drops-200375}, doi = {10.4230/LIPIcs.SoCG.2024.92}, annote = {Keywords: Hilbert metric, Macbeath Regions, Polar Bodies, Convexity} }

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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of n points in the Hilbert geometry defined by a convex m-gon. The algorithm runs in O(n (log n + log³ m)) expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time O(n h log² m), where h is the number of points on the hull’s boundary.

Auguste H. Gezalyan, Soo H. Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, and David M. Mount. Delaunay Triangulations in the Hilbert Metric. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gezalyan_et_al:LIPIcs.SWAT.2024.25, author = {Gezalyan, Auguste H. and Kim, Soo H. and Lopez, Carlos and Skora, Daniel and Stefankovic, Zofia and Mount, David M.}, title = {{Delaunay Triangulations in the Hilbert Metric}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.25}, URN = {urn:nbn:de:0030-drops-200657}, doi = {10.4230/LIPIcs.SWAT.2024.25}, annote = {Keywords: Delaunay Triangulations, Hilbert metric, convexity, randomized algorithms} }

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

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities.
In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation.
We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in ℝ^d from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute ε-approximation to this query in time O(log (1/ε)) using O(1/ε^{d/2}) storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.

Ahmed Abdelkader and David M. Mount. Smooth Distance Approximation. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abdelkader_et_al:LIPIcs.ESA.2023.5, author = {Abdelkader, Ahmed and Mount, David M.}, title = {{Smooth Distance Approximation}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {5:1--5:18}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.5}, URN = {urn:nbn:de:0030-drops-186589}, doi = {10.4230/LIPIcs.ESA.2023.5}, annote = {Keywords: Approximation algorithms, convexity, continuity, partition of unity} }

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

Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body K of diameter Δ in ℝ^d for fixed d. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that Θ((Δ/ε)^{(d-1)/2}) vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies.
A natural way to characterize a convex object’s skinniness is in terms of its relationship to the Euclidean ball. Given a convex body K, define its volume diameter Δ_d to be the diameter of a Euclidean ball of the same volume as K, and define its surface diameter Δ_{d-1} analogously for surface area. It follows from generalizations of the isoperimetric inequality that Δ ≥ Δ_{d-1} ≥ Δ_d.
Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to O((Δ_{d-1}/ε)^{(d-1)/2}). In this paper, we strengthen this by proving the existence of an approximation with O((Δ_d/ε)^{(d-1)/2}) facets.
This improvement is a result of the combination of a number of new ideas. As in prior work, we exploit properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the following more general problem. Given two convex bodies, one nested within the other, find a low-complexity convex polytope that is sandwiched between them. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.

Sunil Arya and David M. Mount. Optimal Volume-Sensitive Bounds for Polytope Approximation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arya_et_al:LIPIcs.SoCG.2023.9, author = {Arya, Sunil and Mount, David M.}, title = {{Optimal Volume-Sensitive Bounds for Polytope Approximation}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {9:1--9:16}, 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.2023.9}, URN = {urn:nbn:de:0030-drops-178592}, doi = {10.4230/LIPIcs.SoCG.2023.9}, annote = {Keywords: Approximation algorithms, convexity, Macbeath regions} }

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

The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any m-sided convex polygon Ω in the plane, we present two randomized incremental algorithms and one deterministic algorithm. The first randomized algorithm and the deterministic algorithm compute the Voronoi diagram of a set of n point sites. The second randomized algorithm extends this to compute the Voronoi diagram of the set of n sites, each of which may be a point or a line segment. Our algorithms all run in expected time O(m n log n). The algorithms use O(m n) storage, which matches the worst-case combinatorial complexity of the Voronoi diagram in the Hilbert metric.

Auguste H. Gezalyan and David M. Mount. Voronoi Diagrams in the Hilbert Metric. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gezalyan_et_al:LIPIcs.SoCG.2023.35, author = {Gezalyan, Auguste H. and Mount, David M.}, title = {{Voronoi Diagrams in the Hilbert Metric}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {35:1--35:16}, 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.2023.35}, URN = {urn:nbn:de:0030-drops-178851}, doi = {10.4230/LIPIcs.SoCG.2023.35}, annote = {Keywords: Voronoi diagrams, Hilbert metric, convexity, randomized algorithms} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

The problem of nearest-neighbor classification is a fundamental technique in machine-learning. Given a training set P of n labeled points in ℝ^d, and an approximation parameter 0 < ε ≤ 1/2, any unlabeled query point should be classified with the class of any of its ε-approximate nearest-neighbors in P. Answering these queries efficiently has been the focus of extensive research, proposing techniques that are mainly tailored towards resolving the more general problem of ε-approximate nearest-neighbor search. While the latest can only hope to provide query time and space complexities dependent on n, the problem of nearest-neighbor classification accepts other parameters more suitable to its analysis. Such is the number k_ε of ε-border points, which describes the complexity of boundaries between sets of points of different classes.
This paper presents a new data structure called Chromatic AVD. This is the first approach for ε-approximate nearest-neighbor classification whose space and query time complexities are only dependent on ε, k_ε and d, while being independent on both n and Δ, the spread of P.

Alejandro Flores-Velazco and David M. Mount. Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{floresvelazco_et_al:LIPIcs.ESA.2021.44, author = {Flores-Velazco, Alejandro and Mount, David M.}, title = {{Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.44}, URN = {urn:nbn:de:0030-drops-146252}, doi = {10.4230/LIPIcs.ESA.2021.44}, annote = {Keywords: approximate nearest-neighbor searching, nearest-neighbor classification, geometric data structures, space-time tradeoffs} }

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

Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider nearest-neighbor queries against a set of line segments in ℝ^d, for constant dimension d. Given a set S of n disjoint line segments in ℝ^d and an error parameter ε > 0, the objective is to build a data structure such that for any query point q, it is possible to return a line segment whose Euclidean distance from q is at most (1+ε) times the distance from q to its nearest line segment. We present a data structure for this problem with storage O((n²/ε^d) log (Δ/ε)) and query time O(log (max(n,Δ)/ε)), where Δ is the spread of the set of segments S. Our approach is based on a covering of space by anisotropic elements, which align themselves according to the orientations of nearby segments.

Ahmed Abdelkader and David M. Mount. Approximate Nearest-Neighbor Search for Line Segments. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abdelkader_et_al:LIPIcs.SoCG.2021.4, author = {Abdelkader, Ahmed and Mount, David M.}, title = {{Approximate Nearest-Neighbor Search for Line Segments}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {4:1--4:15}, 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.2021.4}, URN = {urn:nbn:de:0030-drops-138039}, doi = {10.4230/LIPIcs.SoCG.2021.4}, annote = {Keywords: Approximate nearest-neighbor searching, Approximate Voronoi diagrams, Line segments, Macbeath regions} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Given a training set P of labeled points, the nearest-neighbor rule predicts the class of an unlabeled query point as the label of its closest point in the set. To improve the time and space complexity of classification, a natural question is how to reduce the training set without significantly affecting the accuracy of the nearest-neighbor rule. Nearest-neighbor condensation deals with finding a subset R ⊆ P such that for every point p ∈ P, p’s nearest-neighbor in R has the same label as p. This relates to the concept of coresets, which can be broadly defined as subsets of the set, such that an exact result on the coreset corresponds to an approximate result on the original set. However, the guarantees of a coreset hold for any query point, and not only for the points of the training set.
This paper introduces the concept of coresets for nearest-neighbor classification. We extend existing criteria used for condensation, and prove sufficient conditions to correctly classify any query point when using these subsets. Additionally, we prove that finding such subsets of minimum cardinality is NP-hard, and propose quadratic-time approximation algorithms with provable upper-bounds on the size of their selected subsets. Moreover, we show how to improve one of these algorithms to have subquadratic runtime, being the first of this kind for condensation.

Alejandro Flores-Velazco and David M. Mount. Coresets for the Nearest-Neighbor Rule. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 47:1-47:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{floresvelazco_et_al:LIPIcs.ESA.2020.47, author = {Flores-Velazco, Alejandro and Mount, David M.}, title = {{Coresets for the Nearest-Neighbor Rule}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {47:1--47:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.47}, URN = {urn:nbn:de:0030-drops-129138}, doi = {10.4230/LIPIcs.ESA.2020.47}, annote = {Keywords: coresets, nearest-neighbor rule, classification, nearest-neighbor condensation, training-set reduction, approximate nearest-neighbor, approximation algorithms} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

As the prevalence of E-commerce continues to grow, the efficient operation of warehouses and fulfillment centers is becoming increasingly important. To this end, many such warehouses are adding automation in order to help streamline operations, drive down costs, and increase overall efficiency. The introduction of automation comes with the opportunity for new theoretical models and computational problems with which to better understand and optimize such systems.
These systems often maintain a warehouse of standardized portable storage units, which are stored and retrieved by robotic workers. In general, there are two principal issues in optimizing such a system: where in the warehouse each storage unit should be located and how best to retrieve them. These two concerns naturally go hand-in-hand, but are further complicated by the unknown request frequencies of stored products. Analogous to virtual-memory systems, the more popular and oft-requested an item is, the more efficient its retrieval should be. In this paper, we propose a theoretical model for organizing portable storage units in a warehouse subject to an online sequence of access requests. We consider two formulations, depending on whether there is a single access point or multiple access points. We present algorithms that are O(1)-competitive with respect to an optimal algorithm. In the case of a single access point, our solution is also asymptotically optimal with respect to density.

Philip Dasler and David M. Mount. Online Algorithms for Warehouse Management. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 56:1-56:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dasler_et_al:LIPIcs.ISAAC.2019.56, author = {Dasler, Philip and Mount, David M.}, title = {{Online Algorithms for Warehouse Management}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {56:1--56:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.56}, URN = {urn:nbn:de:0030-drops-115526}, doi = {10.4230/LIPIcs.ISAAC.2019.56}, annote = {Keywords: Warehouse management, online algorithms, competitive analysis, robotics} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Approximation problems involving a single convex body in R^d have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions d <= 3 and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter epsilon > 0, we show how to independently preprocess two polytopes A,B subset R^d into data structures of size O(1/epsilon^{(d-1)/2}) such that we can answer in polylogarithmic time whether A and B intersect approximately. More generally, we can answer this for the images of A and B under affine transformations. Next, we show how to epsilon-approximate the Minkowski sum of two given polytopes defined as the intersection of n halfspaces in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to epsilon-approximate the width of a set of n points in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0, a major improvement over the previous bound of roughly O(n + 1/epsilon^{d-1}) time.

Sunil Arya, Guilherme D. da Fonseca, and David M. Mount. Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arya_et_al:LIPIcs.ESA.2018.3, author = {Arya, Sunil and da Fonseca, Guilherme D. and Mount, David M.}, title = {{Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.3}, URN = {urn:nbn:de:0030-drops-94664}, doi = {10.4230/LIPIcs.ESA.2018.3}, annote = {Keywords: Minkowski sum, convex intersection, width, approximation} }

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

Convex bodies are ubiquitous in computational geometry and optimization theory. The high combinatorial complexity of multidimensional convex polytopes has motivated the development of algorithms and data structures for approximate representations. This paper demonstrates an intriguing connection between convex approximation and the classical concept of Delone sets from the theory of metric spaces. It shows that with the help of a classical structure from convexity theory, called a Macbeath region, it is possible to construct an epsilon-approximation of any convex body as the union of O(1/epsilon^{(d-1)/2}) ellipsoids, where the center points of these ellipsoids form a Delone set in the Hilbert metric associated with the convex body. Furthermore, a hierarchy of such approximations yields a data structure that answers epsilon-approximate polytope membership queries in O(log (1/epsilon)) time. This matches the best asymptotic results for this problem, by a data structure that both is simpler and arguably more elegant.

Ahmed Abdelkader and David M. Mount. Economical Delone Sets for Approximating Convex Bodies. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abdelkader_et_al:LIPIcs.SWAT.2018.4, author = {Abdelkader, Ahmed and Mount, David M.}, title = {{Economical Delone Sets for Approximating Convex Bodies}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {4:1--4:12}, 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.2018.4}, URN = {urn:nbn:de:0030-drops-88300}, doi = {10.4230/LIPIcs.SWAT.2018.4}, annote = {Keywords: Approximate polytope membership, Macbeath regions, Delone sets, Hilbert geometry} }

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

The computation of (i) eps-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In each case the input is a set of points in d-dimensional space for a constant d and an approximation parameter eps > 0. In this paper, we describe new algorithms for these problems, achieving significant improvements to the exponent of the eps-dependency in their running times, from roughly d to d/2 for the first two problems and from roughly d/3 to d/4 for problem (iii).
These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decomposed the space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures for (iv) approximate nearest neighbor searching, (v) directional width queries, and (vi) polytope membership queries.

Sunil Arya, Guilherme D. da Fonseca, and David M. Mount. Near-Optimal epsilon-Kernel Construction and Related Problems. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arya_et_al:LIPIcs.SoCG.2017.10, author = {Arya, Sunil and da Fonseca, Guilherme D. and Mount, David M.}, title = {{Near-Optimal epsilon-Kernel Construction and Related Problems}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {10:1--10:15}, 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.2017.10}, URN = {urn:nbn:de:0030-drops-72257}, doi = {10.4230/LIPIcs.SoCG.2017.10}, annote = {Keywords: Approximation, diameter, kernel, coreset, nearest neighbor, polytope membership, bichromatic closest pair, Macbeath regions} }

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

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter $diam(K)$ is given in Euclidean d-dimensional space, where $d$ is a constant. Given an error parameter eps > 0, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most eps diam(K). By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that O(1/eps^{(d-1)/2}) facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is O-tilde(1/eps^{(d-1)/2}), where O-tilde conceals a polylogarithmic factor in 1/eps. This is an improvement upon the best known bound, which is roughly O(1/eps^{d-2}).
Our result is based on a novel combination of both new and old ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Barany and Larman's economical cap covering, which may be of independent interest. Finally, we use a deterministic variation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

Sunil Arya, Guilherme D. da Fonseca, and David M. Mount. On the Combinatorial Complexity of Approximating Polytopes. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arya_et_al:LIPIcs.SoCG.2016.11, author = {Arya, Sunil and da Fonseca, Guilherme D. and Mount, David M.}, title = {{On the Combinatorial Complexity of Approximating Polytopes}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {11:1--11:15}, 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.2016.11}, URN = {urn:nbn:de:0030-drops-59034}, doi = {10.4230/LIPIcs.SoCG.2016.11}, annote = {Keywords: Polytope approximation, Convex polytopes, Macbeath regions} }

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

We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following:
(A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set.
(B) One can decide if a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.

Sariel Har-Peled, Nirman Kumar, David M. Mount, and Benjamin Raichel. Space Exploration via Proximity Search. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 374-389, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{harpeled_et_al:LIPIcs.SOCG.2015.374, author = {Har-Peled, Sariel and Kumar, Nirman and Mount, David M. and Raichel, Benjamin}, title = {{Space Exploration via Proximity Search}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {374--389}, 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.2015.374}, URN = {urn:nbn:de:0030-drops-51004}, doi = {10.4230/LIPIcs.SOCG.2015.374}, annote = {Keywords: Proximity search, implicit point set, probing} }

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

Range searching is a widely-used method in computational geometry for efficiently accessing local regions of a large data set. Typically, range searching involves either counting or reporting the points lying within a given query region, but it is often desirable to compute statistics that better describe the structure of the point set lying within the region, not just the count.
In this paper we consider the geometric minimum spanning tree (MST) problem in the context of range searching where approximation is allowed. We are given a set P of n points in R^d. The objective is to preprocess P so that given an admissible query region Q, it is possible to efficiently approximate the weight of the minimum spanning tree of the subset of P lying within Q. There are two natural sources of approximation error, first by treating Q as a fuzzy object and second by approximating the MST weight itself. To model this, we assume that we are given two positive real approximation parameters eps_q and eps_w. Following the typical practice in approximate range searching, the range is expressed as two shapes Q^- and Q^+, where Q^- is contained in Q which is contained in Q^+, and their boundaries are separated by a distance of at least eps_q diam(Q). Points within Q^- must be included and points external to Q^+ cannot be included. A weight W is a valid answer to the query if there exist subsets P' and P'' of P, such that Q^- is contained in P' which is contained in P'' which is contained in Q^+ and wt(MST(P')) <= W <= (1+eps_w) wt(MST(P'')).
In this paper, we present an efficient data structure for answering such queries. Our approach uses simple data structures based on quadtrees, and it can be applied whenever Q^- and Q^+ are compact sets of constant combinatorial complexity. It uses space O(n), and it answers queries in time O(log n + 1/(eps_q eps_w)^{d + O(1)}). The O(1) term is a small constant independent of dimension, and the hidden constant factor in the overall running time depends on d, but not on eps_q or eps_w. Preprocessing requires knowledge of eps_w, but not eps_q.

Sunil Arya, David M. Mount, and Eunhui Park. Approximate Geometric MST Range Queries. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 781-795, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{arya_et_al:LIPIcs.SOCG.2015.781, author = {Arya, Sunil and Mount, David M. and Park, Eunhui}, title = {{Approximate Geometric MST Range Queries}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {781--795}, 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.2015.781}, URN = {urn:nbn:de:0030-drops-51233}, doi = {10.4230/LIPIcs.SOCG.2015.781}, annote = {Keywords: Geometric data structures, Minimum spanning trees, Range searching, Approximation algorithms} }