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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R' that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm.
On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log n log log n)-approximation algorithm for this problem.

Pankaj K. Agarwal, Tzvika Geft, Dan Halperin, and Erin Taylor. Multi-Robot Motion Planning for Unit Discs with Revolving Areas. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agarwal_et_al:LIPIcs.ISAAC.2022.35, author = {Agarwal, Pankaj K. and Geft, Tzvika and Halperin, Dan and Taylor, Erin}, title = {{Multi-Robot Motion Planning for Unit Discs with Revolving Areas}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {35:1--35:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.35}, URN = {urn:nbn:de:0030-drops-173204}, doi = {10.4230/LIPIcs.ISAAC.2022.35}, annote = {Keywords: motion planning, optimal motion planning, approximation, complexity, NP-hardness} }

Document

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

We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end.
If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12, author = {Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn}, title = {{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12}, URN = {urn:nbn:de:0030-drops-160203}, doi = {10.4230/LIPIcs.SoCG.2022.12}, annote = {Keywords: motion planning, computational geometry, simple polygon} }

Document

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

We study several variants of the problem of moving a convex polytope K, with n edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically:
ii) We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms for those variants, which run in time close to O(n^{8/3}).
iii) We consider the case of a gate (an unbounded window with two parallel infinite edges), and show that K can pass through such a window, by any collision-free rigid motion, iff it can slide through it, an observation that leads to an efficient algorithm for this variant too.
iv) We consider arbitrary compact convex windows, and show that if K can pass through such a window W (by any motion) then K can slide through a slab of width equal to the diameter of W.
v) We show that if a purely translational motion for K through a rectangular window W exists, then K can also slide through W keeping the same orientation as in the translational motion. For a given fixed orientation of K we can determine in linear time whether K can translate (and hence slide) through W keeping the given orientation, and if so plan the motion, also in linear time.
vi) We give an example of a polytope that cannot pass through a certain window by translations only, but can do so when rotations are allowed.
vii) We study the case of a circular window W, and show that, for the regular tetrahedron K of edge length 1, there are two thresholds 1 > δ₁≈ 0.901388 > δ₂≈ 0.895611, such that (a) K can slide through W if the diameter d of W is ≥ 1, (b) K cannot slide through W but can pass through it by a purely translational motion when δ₁ ≤ d < 1, (c) K cannot pass through W by a purely translational motion but can do it when rotations are allowed when δ₂ ≤ d < δ₁, and (d) K cannot pass through W at all when d < δ₂.
viii) Finally, we explore the general setup, where we want to plan a general motion (with all six degrees of freedom) for K through a rectangular window W, and present an efficient algorithm for this problem, with running time close to O(n⁴).

Dan Halperin, Micha Sharir, and Itay Yehuda. Throwing a Sofa Through the Window. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{halperin_et_al:LIPIcs.SoCG.2021.41, author = {Halperin, Dan and Sharir, Micha and Yehuda, Itay}, title = {{Throwing a Sofa Through the Window}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.41}, URN = {urn:nbn:de:0030-drops-138409}, doi = {10.4230/LIPIcs.SoCG.2021.41}, annote = {Keywords: Motion planning, Convex polytopes in 3D} }

Document

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

We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of n unit discs under insertions in O(k log^2 n) update time and O(n) space, where k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has O(log^2 n) update time and O(log n) vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in O(log n) time, using tentative binary search; the lower envelopes are special in that at x=-infty any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with O(n log n) preprocessing time, O(n^{1/2+epsilon} + l) query time and O(log^2 n) amortized update time, where l is the size of the output and for any epsilon>0. The structure requires O(n) storage space.

Pankaj K. Agarwal, Ravid Cohen, Dan Halperin, and Wolfgang Mulzer. Maintaining the Union of Unit Discs Under Insertions with Near-Optimal Overhead. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.26, author = {Agarwal, Pankaj K. and Cohen, Ravid and Halperin, Dan and Mulzer, Wolfgang}, title = {{Maintaining the Union of Unit Discs Under Insertions with Near-Optimal Overhead}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {26:1--26:15}, 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.2019.26}, URN = {urn:nbn:de:0030-drops-104307}, doi = {10.4230/LIPIcs.SoCG.2019.26}, annote = {Keywords: lower envelopes, pseudo-lines, unit discs, range search, dynamic algorithms, tentative binary search} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We describe a general probabilistic framework to address a variety of Fréchet-distance optimization problems. Specifically, we are interested in finding minimal bottleneck-paths in d-dimensional Euclidean space between given start and goal points, namely paths that minimize the maximal value over a continuous cost map. We present an efficient and simple sampling-based framework for this problem, which is inspired by, and draws ideas from, techniques for robot motion planning. We extend the framework to handle not only standard bottleneck pathfinding, but also the more demanding case, where the path needs to be monotone in all dimensions. Finally, we provide experimental results of the framework on several types of problems.

Kiril Solovey and Dan Halperin. Sampling-Based Bottleneck Pathfinding with Applications to Fréchet Matching. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{solovey_et_al:LIPIcs.ESA.2016.76, author = {Solovey, Kiril and Halperin, Dan}, title = {{Sampling-Based Bottleneck Pathfinding with Applications to Fr\'{e}chet Matching}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {76:1--76:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.76}, URN = {urn:nbn:de:0030-drops-64179}, doi = {10.4230/LIPIcs.ESA.2016.76}, annote = {Keywords: Computational geometry, Fr\'{e}chet distances, sampling-based algorithms, random geometric graphs, bottleneck pathfinding} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Dan Halperin and Günter Rote. Computational Geometry (Dagstuhl Seminar 03121). Dagstuhl Seminar Report 372, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2003)

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@TechReport{halperin_et_al:DagSemRep.372, author = {Halperin, Dan and Rote, G\"{u}nter}, title = {{Computational Geometry (Dagstuhl Seminar 03121)}}, pages = {1--6}, ISSN = {1619-0203}, year = {2003}, type = {Dagstuhl Seminar Report}, number = {372}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.372}, URN = {urn:nbn:de:0030-drops-152529}, doi = {10.4230/DagSemRep.372}, }