35 Search Results for "A. Akitaya, Hugo"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.20
The Spanning Ratio of the Directed Θ₆-Graph Is 5

Authors: Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.
Cite as

Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2026.20,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John and Hill, Darryl},
  title =	{{The Spanning Ratio of the Directed \Theta₆-Graph Is 5}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.20},
  URN =		{urn:nbn:de:0030-drops-258268},
  doi =		{10.4230/LIPIcs.SoCG.2026.20},
  annote =	{Keywords: Geometric Spanners, Theta Graphs, Directed Theta Graphs, Spanning Ratio, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.36
Upward Book Embeddings of Partitioned Digraphs

Authors: Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph G = (V, ⋃^k_{i=1} E_i), that is, a digraph whose edge set is partitioned into k subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput. 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for k = 1 and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for k ≥ 3. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case k = 2. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when k = 2, thus closing the complexity gap for the problem. Second, we show that, for an n-vertex partitioned digraph with a prescribed planar embedding, the existence of an upward book embedding that respects the given planar embedding can be tested in O(n log³ n) time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial 2-trees.
Cite as

Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter. Upward Book Embeddings of Partitioned Digraphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dalozzo_et_al:LIPIcs.SoCG.2026.36,
  author =	{Da Lozzo, Giordano and Frati, Fabrizio and Rutter, Ignaz},
  title =	{{Upward Book Embeddings of Partitioned Digraphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.36},
  URN =		{urn:nbn:de:0030-drops-258424},
  doi =		{10.4230/LIPIcs.SoCG.2026.36},
  annote =	{Keywords: upward book embeddings, partitioned digraphs, SPQ-trees, 2-trees}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.96
Sliding Cubes in Parallel (Media Exposition)

Authors: Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The sliding cubes model serves as a well-established theoretical framework for formalizing and analyzing reconfiguration algorithms in modular robotic systems built from face-connected cubic modules. We extend the parallel sliding cubes model from two to three dimensions, presenting new algorithms, surprising complexity results, and a generalization of the best known bounds from two to three dimensions. A companion video visualizes and explains our results.
Cite as

Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock. Sliding Cubes in Parallel (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 96:1-96:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{a.akitaya_et_al:LIPIcs.SoCG.2026.96,
  author =	{A. Akitaya, Hugo and Dorfer, Joseph and Kramer, Peter and Rieck, Christian and Samanta, Soham and Shahrouzi, Gabriel and Stock, Frederick},
  title =	{{Sliding Cubes in Parallel}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{96:1--96:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.96},
  URN =		{urn:nbn:de:0030-drops-259020},
  doi =		{10.4230/LIPIcs.SoCG.2026.96},
  annote =	{Keywords: Sliding squares, parallel motion, reconfigurability, three dimensions, constant makespan, log-APX hardness, NP-hardness, worst-case optimality}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.97
"Visualizing" the CG Community (Media Exposition)

Authors: Oswin Aichholzer, Hugo A. Akitaya, Anna Brötzner, Peter Kramer, Christian Rieck, and Frederick Stock

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We analyze and visualize collaboration within the Computational Geometry community by modeling co-authorship relations as a graph, where nodes correspond to individual researchers and edges represent shared publications. By aggregating and time-slicing conference data, we construct a dynamic representation of the community that supports both interactive visualization and structured search.
Cite as

Oswin Aichholzer, Hugo A. Akitaya, Anna Brötzner, Peter Kramer, Christian Rieck, and Frederick Stock. "Visualizing" the CG Community (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 97:1-97:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2026.97,
  author =	{Aichholzer, Oswin and A. Akitaya, Hugo and Br\"{o}tzner, Anna and Kramer, Peter and Rieck, Christian and Stock, Frederick},
  title =	{{"Visualizing" the CG Community}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{97:1--97:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.97},
  URN =		{urn:nbn:de:0030-drops-259039},
  doi =		{10.4230/LIPIcs.SoCG.2026.97},
  annote =	{Keywords: CG community, visualization, graph parameters, web application}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.104
Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition)

Authors: Soham Samanta, Hugo A. Akitaya, Erik Demaine, and Martin Demaine

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
This paper introduces interactive software tools for studying 2-face path unfoldings of the tesseract (4D hypercube). We present: (1) an algorithm to verify whether a given 24-omino is a valid path unfolding of the tesseract, (2) a web-based visualization tool for exploring and animating unfolding sequences with smooth 3D interpolation, and (3) a design interface integrated with SVG Painter, with a similar design to Demaine’s SVG Painter, to create custom unfoldings. We demonstrate these tools by designing a geometric font of 36 path unfoldings resembling Latin letters and digits, illustrating the rich diversity and accessibility of tesseract geometry.
Cite as

Soham Samanta, Hugo A. Akitaya, Erik Demaine, and Martin Demaine. Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 104:1-104:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{samanta_et_al:LIPIcs.SoCG.2026.104,
  author =	{Samanta, Soham and A. Akitaya, Hugo and Demaine, Erik and Demaine, Martin},
  title =	{{Interactive Visualization and Verification Tools for Tesseract Path Unfoldings}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{104:1--104:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.104},
  URN =		{urn:nbn:de:0030-drops-259104},
  doi =		{10.4230/LIPIcs.SoCG.2026.104},
  annote =	{Keywords: unfolding, polyominoes, tesseract, path unfolding, visualization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.44
Tilt Automata: Gathering Particles with Uniform External Control

Authors: Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As a corollary, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.
Cite as

Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer. Tilt Automata: Gathering Particles with Uniform External Control. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.44,
  author =	{Fekete, S\'{a}ndor P. and Friemel, Jonas and Kramer, Peter and Reinhardt, Jan-Marc and Rieck, Christian and Scheffer, Christian},
  title =	{{Tilt Automata: Gathering Particles with Uniform External Control}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{44:1--44:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.44},
  URN =		{urn:nbn:de:0030-drops-258508},
  doi =		{10.4230/LIPIcs.SoCG.2026.44},
  annote =	{Keywords: Uniform control, gathering, full tilt, polyominoes, synchronizing automata}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.14
General Computation Using Slidable Tiles with Deterministic Global Forces

Authors: Alberto Avila-Jimenez, David Barreda, Sarah-Laurie Evans, Austin Luchsinger, Aiden Massie, Robert Schweller, Evan Tomai, and Tim Wylie

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We study the computational power of the Full-Tilt model of motion planning, where slidable polyominos are moved maximally around a board by way of a sequence of directional "tilts." We focus on the deterministic scenario in which the tilts constitute a repeated clockwise rotation. We show that general-purpose computation is possible within this framework by providing a direct and efficient simulation of space-bounded Turing machines in which one computational step of the machine is simulated per O(1) rotations. We further show that the initial tape of the machine can be programmed by an initial tilt-sequence preceding the rotations. This result immediately implies new PSPACE-completeness results for the well-studied problems of occupancy (deciding if a given board location can be occupied by a tile), vacancy (deciding if a location can be emptied), relocation (deciding if a tile can be moved from one location to another), and reconfiguration (can a given board configuration be reconfigured into a second given configuration) that hold even for deterministically repeating tilt cycles such as rotations. All of our PSPACE-completeness results hold even when there is only a single domino in the system beyond singleton tiles. Following, we show that these results work in the Single-Step tilt model for larger constant cycles. We then investigate computational efficiency by showing a modification to implement a two-tape Turing machine in the Full-Tilt model and Systolic Arrays in the Single-Step model. Finally, we show a cyclic implementation for tilt-efficient Threshold Circuits.
Cite as

Alberto Avila-Jimenez, David Barreda, Sarah-Laurie Evans, Austin Luchsinger, Aiden Massie, Robert Schweller, Evan Tomai, and Tim Wylie. General Computation Using Slidable Tiles with Deterministic Global Forces. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 14:1-14:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{avilajimenez_et_al:LIPIcs.ITCS.2026.14,
  author =	{Avila-Jimenez, Alberto and Barreda, David and Evans, Sarah-Laurie and Luchsinger, Austin and Massie, Aiden and Schweller, Robert and Tomai, Evan and Wylie, Tim},
  title =	{{General Computation Using Slidable Tiles with Deterministic Global Forces}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{14:1--14:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.14},
  URN =		{urn:nbn:de:0030-drops-253019},
  doi =		{10.4230/LIPIcs.ITCS.2026.14},
  annote =	{Keywords: motion planning, global control, external forces, deterministic computation, occupancy, vacancy}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.23
The Price of Connectivity Augmentation on Planar Graphs

Authors: Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, and Torsten Ueckerdt

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Given two classes of graphs, 𝒢₁ ⊆ 𝒢₂, and a c-connected graph G ∈ 𝒢₁, we wish to augment G with a smallest cardinality set of new edges F to obtain a k-connected graph G' = (V,E∪ F) ∈ 𝒢₂. In general, this is the c → k connectivity augmentation problem. Previous research considered variants where 𝒢₁ = 𝒢₂ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the c → k augmentation problem is NP-complete when 2 ≤ c < k ≤ 5. However, the connectivity of the augmented graph G' is at most 5 if 𝒢₂ is limited to planar graphs. We initiate the study of the c → k connectivity augmentation problem for arbitrary k ∈ ℕ, where 𝒢₁ is the class of planar graphs, plane graphs, or planar straight-line graphs, and 𝒢₂ is a beyond-planar class of graphs: 𝓁-planar, 𝓁-plane topological, or 𝓁-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity k and the local crossing number 𝓁 of the augmented graph G'. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.
Cite as

Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, and Torsten Ueckerdt. The Price of Connectivity Augmentation on Planar Graphs. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.GD.2025.23,
  author =	{A. Akitaya, Hugo and Dallant, Justin and Demaine, Erik D. and Kaufmann, Michael and Kleist, Linda and Stock, Frederick and T\'{o}th, Csaba D. and Ueckerdt, Torsten},
  title =	{{The Price of Connectivity Augmentation on Planar Graphs}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{23:1--23:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.23},
  URN =		{urn:nbn:de:0030-drops-250095},
  doi =		{10.4230/LIPIcs.GD.2025.23},
  annote =	{Keywords: connectivity augmentation, local crossing number, flip distance}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.28
Sliding Squares in Parallel

Authors: Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given n square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of sliding moves to transform it into a desired goal configuration, maintaining connectivity of the configuration at all times. Recent work has aimed at minimizing the total number of moves, resulting in fully sequential schedules that can perform reconfiguration in 𝒪(n²) moves, or 𝒪(nP) for arrangements of bounding box perimeter size P. We provide first results in the sliding square model that exploit parallel motion, performing reconfiguration in worst-case optimal makespan of 𝒪(P). We also provide tight bounds on the complexity of the problem by showing that even deciding the possibility of reconfiguration within makespan 1 is NP-complete in the unlabeled case. In the labeled variant, we note that deciding the same for makespan 2 is NP-complete, while makespan 1 is straightforward.
Cite as

Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner. Sliding Squares in Parallel. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2025.28,
  author =	{A. Akitaya, Hugo and Fekete, S\'{a}ndor P. and Kramer, Peter and Molaei, Saba and Rieck, Christian and Stock, Frederick and Wallner, Tobias},
  title =	{{Sliding Squares in Parallel}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.28},
  URN =		{urn:nbn:de:0030-drops-244961},
  doi =		{10.4230/LIPIcs.ESA.2025.28},
  annote =	{Keywords: Sliding squares, parallel motion, reconfigurability, motion planning, multi-agent path finding, makespan, swarm robotics, computational geometry}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.75
An Improved Bound for Plane Covering Paths

Authors: Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, and Leonidas Theocharous

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
A covering path for a finite set P of points in the plane is a polygonal path such that every point of P lies on a segment of the path. The vertices of the path need not be at points of P. A covering path is plane if its segments do not cross each other. Let π(n) be the minimum number such that every set of n points in the plane admits a plane covering path with at most π(n) segments. We prove that π(n) ≤ ⌈6n/7⌉. This improves the previous best-known upper bound of ⌈21n/22⌉, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple O(n log n)-time algorithm for computing a plane covering path.
Cite as

Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, and Leonidas Theocharous. An Improved Bound for Plane Covering Paths. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 75:1-75:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2025.75,
  author =	{A. Akitaya, Hugo and Aloupis, Greg and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Gavoille, Cyril and Iacono, John and Kleist, Linda and Smid, Michiel and Souvaine, Diane and Theocharous, Leonidas},
  title =	{{An Improved Bound for Plane Covering Paths}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{75:1--75:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.75},
  URN =		{urn:nbn:de:0030-drops-245432},
  doi =		{10.4230/LIPIcs.ESA.2025.75},
  annote =	{Keywords: Covering Path, Upper Bound, Simple Algorithm}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.12
Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better

Authors: Jacobus Conradi and Anne Driemel

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Many application areas collect unstructured trajectory data. In subtrajectory clustering, one is interested to find patterns in this data using a hybrid combination of segmentation and clustering. We analyze two variants of this problem based on the well-known SetCover and CoverageMaximization problems. In both variants the set system is induced by metric balls under the Fréchet distance centered at polygonal curves. Our algorithms focus on improving the running time of the update step of the generic greedy algorithm by means of a careful combination of sweeps through a candidate space. In the first variant, we are given a polygonal curve P of complexity n, distance threshold Δ and complexity bound 𝓁 and the goal is to identify a minimum-size set of center curves 𝒞, where each center curve is of complexity at most 𝓁 and every point p on P is covered. A point p on P is covered if it is part of a subtrajectory π_p of P such that there is a center c ∈ 𝒞 whose Fréchet distance to π_p is at most Δ. We present an approximation algorithm for this problem with a running time of 𝒪((n²𝓁 + √{k_Δ}n^{5/2})log²n), where k_Δ is the size of an optimal solution. The algorithm gives a bicriterial approximation guarantee that relaxes the Fréchet distance threshold by a constant factor and the size of the solution by a factor of 𝒪(log n). The second problem variant asks for the maximum fraction of the input curve P that can be covered using k center curves, where k ≤ n is a parameter to the algorithm. For the second problem variant, our techniques lead to an algorithm with a running time of 𝒪((k+𝓁)n²log²n) and similar approximation guarantees. Note that in both algorithms k,k_Δ ∈ O(n) and hence the running time is cubic, or better if k ≪ n.
Cite as

Jacobus Conradi and Anne Driemel. Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{conradi_et_al:LIPIcs.ESA.2025.12,
  author =	{Conradi, Jacobus and Driemel, Anne},
  title =	{{Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.12},
  URN =		{urn:nbn:de:0030-drops-244806},
  doi =		{10.4230/LIPIcs.ESA.2025.12},
  annote =	{Keywords: Clustering, Set cover, Fr\'{e}chet distance, Approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.9
Online Routing in Directed Yao₄^∞ Graphs

Authors: Prosenjit Bose, Jean-Lou De Carufel, and John Stuart

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
The x⃑{Yao₄^∞} and x⃑{Yao₄} graphs are two families of directed geometric graphs whose vertices are points in the plane, and each vertex has up to four outgoing edges. Consider a horizontal and a vertical line through each vertex v, defining four quadrants around v. Then v has an outgoing edge to the closest vertex in each of its four quadrants. When the distance is measured using the Euclidean norm, the resulting graph is the x⃑{Yao₄} graph, whereas with the L_∞-norm, we obtain the x⃑{Yao^{∞}₄} graph, which is a sub-graph of the well-known L_∞-Delaunay graph. In this paper, we provide a local routing algorithm with routing ratio at most 85.22 for x⃑{Yao^{∞}₄} graphs. Prior to this work, no constant spanning or routing ratios for x⃑{Yao₄^∞} graphs were previously known. Now, x⃑{Yao₄^∞} graphs are the sparsest family of directed planar graphs supporting a competitive local routing strategy. Furthermore, we show that no local routing algorithm for x⃑{Yao₄^∞} graphs can have a routing ratio lower than 7+√2≈ 8.41. Moreover, we prove that the spanning ratio is at least 5+√2≈ 6.41 in the worst case. The techniques we develop in this paper also allow us to prove lower bounds of 7-√3+√{5-2√3}≈ 6.51 and 7+√2 for the spanning and routing ratios of x⃑{Yao₄}, respectively.
Cite as

Prosenjit Bose, Jean-Lou De Carufel, and John Stuart. Online Routing in Directed Yao₄^∞ Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.WADS.2025.9,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John},
  title =	{{Online Routing in Directed Yao₄^∞ Graphs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.9},
  URN =		{urn:nbn:de:0030-drops-242404},
  doi =		{10.4230/LIPIcs.WADS.2025.9},
  annote =	{Keywords: Geometric Spanners, Yao Graphs, Local Routing Algorithms}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.10
On Geodesic Disks Enclosing Many Points

Authors: Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. Silveira, and Tyler Tuttle

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Let Π(n) be the largest number such that for every set S of n points in a polygon P, there always exist two points x, y ∈ S, where every geodesic disk containing x and y contains Π(n) points of S. We establish upper and lower bounds for Π(n), and show that ⌈n/5⌉ +1 ≤ Π(n) ≤ ⌈n/4⌉ +1. We also show that there always exist two points x, y ∈ S such that every geodesic disk with x and y on its boundary contains at least 16/665(n-2) ≈ ⌈(n-2)/41.6⌉ points both inside and outside the disk. For the special case where the points of S are restricted to be the vertices of a geodesically convex polygon we give a tight bound of ⌈n/3⌉ + 1. We provide the same tight bound when we only consider geodesic disks having x and y as diametral endpoints. Finally, we give a lower bound of ⌈(n-2)/36⌉+2 for the two-colored version of the problem.
Cite as

Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. Silveira, and Tyler Tuttle. On Geodesic Disks Enclosing Many Points. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.WADS.2025.10,
  author =	{Bose, Prosenjit and Esteban, Guillermo and Orden, David and Silveira, Rodrigo I. and Tuttle, Tyler},
  title =	{{On Geodesic Disks Enclosing Many Points}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.10},
  URN =		{urn:nbn:de:0030-drops-242414},
  doi =		{10.4230/LIPIcs.WADS.2025.10},
  annote =	{Keywords: Enclosing disks, Geodesic disks, Bichromatic}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.11
Crossing and Independent Families Among Polygons

Authors: Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Given a set A of points in the plane, a family of line segments forming a matching in A is called crossing (or independent) if each pair of segments in the family intersects (or is non-intersecting, respectively). In past works, these notions have been generalized to polygons by identifying the points in A with the vertices of a given set of polygons and forbidding the line segments from intersecting or overlapping with polygon walls. In this work, we study the computational complexity of computing maximum crossing and independent families in this more general setting. As our first two results, we show that both problems are NP-hard already when the polygons are triangles. Motivated by this, we turn to parameterized algorithms. For our main algorithmic results, we consider the number of polygons on the input as the natural parameter and under this parameterization obtain a fixed-parameter algorithm for computing a largest crossing family among these polygons, and a separate XP-algorithm for computing a largest independent family that lies in one of the faces of the polygonal domain.
Cite as

Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada. Crossing and Independent Families Among Polygons. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brotzner_et_al:LIPIcs.WADS.2025.11,
  author =	{Br\"{o}tzner, Anna and Ganian, Robert and Hamm, Thekla and Klute, Fabian and Parada, Irene},
  title =	{{Crossing and Independent Families Among Polygons}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.11},
  URN =		{urn:nbn:de:0030-drops-242424},
  doi =		{10.4230/LIPIcs.WADS.2025.11},
  annote =	{Keywords: crossing families, crossing-free matchings, segment intersection graphs, computational geometry, parameterized algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.133
Faster, Deterministic and Space Efficient Subtrajectory Clustering

Authors: Ivor van der Hoog, Thijs van der Horst, and Tim Ophelders

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Given a trajectory T and a distance Δ, we wish to find a set C of curves of complexity at most 𝓁, such that we can cover T with subcurves that each are within Fréchet distance Δ to at least one curve in C. We call C an (𝓁,Δ)-clustering and aim to find an (𝓁,Δ)-clustering of minimum cardinality. This problem variant was introduced by Akitaya et al. (2021) and shown to be NP-complete. The main focus has therefore been on bicriteria approximation algorithms, allowing for the clustering to be an (𝓁, Θ(Δ))-clustering of roughly optimal size. We present algorithms that construct (𝓁,4Δ)-clusterings of 𝒪(k log n) size, where k is the size of the optimal (𝓁, Δ)-clustering. We use 𝒪(n³) space and 𝒪(k n³ log⁴ n) time. Our algorithms significantly improve upon the clustering quality (improving the approximation factor in Δ) and size (whenever 𝓁 ∈ Ω(log n / log k)). We offer deterministic running times improving known expected bounds by a factor near-linear in 𝓁. Additionally, we match the space usage of prior work, and improve it substantially, by a factor super-linear in n𝓁, when compared to deterministic results.
Cite as

Ivor van der Hoog, Thijs van der Horst, and Tim Ophelders. Faster, Deterministic and Space Efficient Subtrajectory Clustering. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.ICALP.2025.133,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Ophelders, Tim},
  title =	{{Faster, Deterministic and Space Efficient Subtrajectory Clustering}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{133:1--133:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.133},
  URN =		{urn:nbn:de:0030-drops-235109},
  doi =		{10.4230/LIPIcs.ICALP.2025.133},
  annote =	{Keywords: Fr\'{e}chet distance, clustering, set cover}
}
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