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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 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} }

Document

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

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from alpha-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an epsilon-sample on the bounding surface, with a weak sigma-sparsity condition, we work with the balls of radius delta times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of epsilon, sigma and delta, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.

Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi. Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abdelkader_et_al:LIPIcs.SoCG.2018.1, author = {Abdelkader, Ahmed and Bajaj, Chandrajit L. and Ebeida, Mohamed S. and Mahmoud, Ahmed H. and Mitchell, Scott A. and Owens, John D. and Rushdi, Ahmad A.}, title = {{Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {1:1--1:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.1}, URN = {urn:nbn:de:0030-drops-87147}, doi = {10.4230/LIPIcs.SoCG.2018.1}, annote = {Keywords: sampling conditions, surface reconstruction, polyhedral meshing, Voronoi} }

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Multimedia Exposition

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

Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.

Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi. VoroCrust Illustrated: Theory and Challenges (Multimedia Exposition). In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 77:1-77:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abdelkader_et_al:LIPIcs.SoCG.2018.77, author = {Abdelkader, Ahmed and Bajaj, Chandrajit L. and Ebeida, Mohamed S. and Mahmoud, Ahmed H. and Mitchell, Scott A. and Owens, John D. and Rushdi, Ahmad A.}, title = {{VoroCrust Illustrated: Theory and Challenges}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {77:1--77:4}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.77}, URN = {urn:nbn:de:0030-drops-87903}, doi = {10.4230/LIPIcs.SoCG.2018.77}, annote = {Keywords: sampling, surface reconstruction, polyhedral meshing, Voronoi} }

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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 49, 8th International Conference on Fun with Algorithms (FUN 2016)

We study the computational complexity of a variant of the popular 2048 game in which no new tiles are generated after each move. As usual, instances are defined on rectangular boards of arbitrary size. We consider the natural decision problems of achieving a given constant tile value, score or number of moves. We also consider approximating the maximum achievable value for these three objectives. We prove all these problems are NP-hard by a reduction from 3SAT.
Furthermore, we consider potential extensions of these results to a similar variant of the Threes! game. To this end, we report on a peculiar motion pattern, that is not possible in 2048, which we found much harder to control by similar board designs.

Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. 2048 Without New Tiles Is Still Hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abdelkader_et_al:LIPIcs.FUN.2016.1, author = {Abdelkader, Ahmed and Acharya, Aditya and Dasler, Philip}, title = {{2048 Without New Tiles Is Still Hard}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {1:1--1:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.1}, URN = {urn:nbn:de:0030-drops-58858}, doi = {10.4230/LIPIcs.FUN.2016.1}, annote = {Keywords: Complexity of Games, 2048} }