Document

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

We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let z(G) be the smallest number of zombies required to catch the survivor on a graph G with n vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that z(G) = Θ(n). We also show that there exist maximum-degree-3 outerplanar graphs such that z(G) = Ω(n/log(n)).
Let z_L(G) be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph G. We show that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that z_L(G) ≤ 2 for connected outerplanar graphs and this bound is tight in the worst case. We show that z_L(G) ≤ k for connected graphs with treedepth k. This result implies that z_L(G) is at most (k+1)log n for connected graphs with treewidth k, O(√n) for connected planar graphs, O(√{gn}) for connected graphs with genus g and O(h√{hn}) for connected graphs with any excluded h-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies.

Prosenjit Bose, Jean-Lou De Carufel, and Thomas Shermer. Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 56:1-56:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bose_et_al:LIPIcs.ISAAC.2022.56, author = {Bose, Prosenjit and De Carufel, Jean-Lou and Shermer, Thomas}, title = {{Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {56:1--56:13}, 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.2022.56}, URN = {urn:nbn:de:0030-drops-173418}, doi = {10.4230/LIPIcs.ISAAC.2022.56}, annote = {Keywords: Pursuit-evasion games, Outerplanar, Graphs, Treedepth, Treewidth} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

The Product Structure Theorem for planar graphs (Dujmović et al. JACM, 67(4):22) states that any planar graph is contained in the strong product of a planar 3-tree, a path, and a 3-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous O(nlog n) time algorithm (Morin. Algorithmica, 85(5):1544-1558).

Prosenjit Bose, Pat Morin, and Saeed Odak. An Optimal Algorithm for Product Structure in Planar Graphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 19:1-19:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2022.19, author = {Bose, Prosenjit and Morin, Pat and Odak, Saeed}, title = {{An Optimal Algorithm for Product Structure in Planar Graphs}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.19}, URN = {urn:nbn:de:0030-drops-161797}, doi = {10.4230/LIPIcs.SWAT.2022.19}, annote = {Keywords: Planar graphs, product structure} }

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Invited Talk

**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

A weighted geometric graph G is a graph whose n vertices are points in the plane and whose m edges are line segments weighted by the Euclidean distance between their endpoints. A t-spanner of G is a connected spanning subgraph G' with the property that for every pair of vertices x, y, the shortest path from x to y in G' has weight at most t ≥ 1 times the shortest path from x to y in G. The parameter t is commonly referred to as the spanning ratio or the stretch factor. Typically, G is a graph with Ω(n²) edges. As such, the goal in this area is to construct a subgraph G' that possesses several desirable properties such as O(n) edges and spanning ratio close to 1. In addition, when planarity is one of the desired properties, variants of Delaunay graphs play a vital role in the construction of planar geometric spanners. In this talk, we will provide a comprehensive overview of various results concerning the spanning ratio, among other several other properties, of different types of Delaunay graphs and their subgraphs.

Prosenjit Bose. Spanning Properties of Variants of the Delaunay Graph (Invited Talk). In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, p. 2:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bose:LIPIcs.ISAAC.2021.2, author = {Bose, Prosenjit}, title = {{Spanning Properties of Variants of the Delaunay Graph}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {2:1--2:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.2}, URN = {urn:nbn:de:0030-drops-154352}, doi = {10.4230/LIPIcs.ISAAC.2021.2}, annote = {Keywords: Delaunay Graph, Geometric Graph, Graph Spanner} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Motivated by the connectivity problem in wireless networks with directional antennas, we study bounded-angle spanning trees. Let P be a set of points in the plane and let α be an angle. An α-ST of P is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ∈ P lie in a wedge of angle α centered at p. We study the following closely related problems for α = 120 degrees (however, our approximation ratios hold for any α ⩾ 120 degrees).
1) The α-minimum spanning tree problem asks for an α-ST of minimum sum of edge lengths. Among many interesting results, Aschner and Katz (ICALP 2014) proved the NP-hardness of this problem and presented a 6-approximation algorithm. Their algorithm finds an α-ST of length at most 6 times the length of the minimum spanning tree (MST). By adopting a somewhat similar approach and using different proof techniques we improve this ratio to 16/3.
2) To examine what is possible with non-uniform wedge angles, we define an ̅α-ST to be a spanning tree with the property that incident edges to all points lie in wedges of average angle α. We present an algorithm to find an ̅α-ST whose largest edge-length and sum of edge lengths are at most 2 and 1.5 times (respectively) those of the MST. These ratios are better than any achievable when all wedges have angle α. Our algorithm runs in linear time after computing the MST.

Ahmad Biniaz, Prosenjit Bose, Anna Lubiw, and Anil Maheshwari. Bounded-Angle Minimum Spanning Trees. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 14:1-14:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biniaz_et_al:LIPIcs.SWAT.2020.14, author = {Biniaz, Ahmad and Bose, Prosenjit and Lubiw, Anna and Maheshwari, Anil}, title = {{Bounded-Angle Minimum Spanning Trees}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {14:1--14:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.14}, URN = {urn:nbn:de:0030-drops-122616}, doi = {10.4230/LIPIcs.SWAT.2020.14}, annote = {Keywords: bounded-angle MST, directional antenna, approximation algorithms} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D₁ and D₂ are disjoint if their intersection is empty (i.e., no interval of D₁ intersects any interval of D₂). There can be three different relations between two disjoint 2-intervals; namely, preceding (<), nested (⊏) and crossing (≬). Two 2-intervals D₁ and D₂ are called R-comparable for some R∈{<,⊏,≬}, if either D₁RD₂ or D₂RD₁. A set 𝒟 of disjoint 2-intervals is ℛ-comparable, for some ℛ⊆{<,⊏,≬} and ℛ≠∅, if every pair of 2-intervals in ℛ are R-comparable for some R∈ℛ. Given a set of 2-intervals and some ℛ⊆{<,⊏,≬}, the objective of the {2-interval pattern problem} is to find a largest subset of 2-intervals that is ℛ-comparable.
The 2-interval pattern problem is known to be W[1]-hard when |ℛ|=3 and NP-hard when |ℛ|=2 (except for ℛ={<,⊏}, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing that it is W[1]-hard for both ℛ={⊏,≬} and ℛ={<,≬} (when parameterized by the size of an optimal solution). This answers the open question posed by Vialette [Encyclopedia of Algorithms, 2008].

Prosenjit Bose, Saeed Mehrabi, and Debajyoti Mondal. Parameterized Complexity of Two-Interval Pattern Problem. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 16:1-16:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2020.16, author = {Bose, Prosenjit and Mehrabi, Saeed and Mondal, Debajyoti}, title = {{Parameterized Complexity of Two-Interval Pattern Problem}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {16:1--16:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.16}, URN = {urn:nbn:de:0030-drops-122630}, doi = {10.4230/LIPIcs.SWAT.2020.16}, annote = {Keywords: Interval graphs, Two-interval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W\lbrack1\rbrack-hardness} }

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

A geometric graph G=(P,E) is a set of points in the plane and edges between pairs of points, where the weight of an edge is equal to the Euclidean distance between its two endpoints. In local routing we find a path through G from a source vertex s to a destination vertex t, using only knowledge of the current vertex, its incident edges, and the locations of s and t. We present an algorithm for local routing on the Delaunay triangulation, and show that it finds a path between a source vertex s and a target vertex t that is not longer than 3.56|st|, improving the previous bound of 5.9|st|.

Nicolas Bonichon, Prosenjit Bose, Jean-Lou De Carufel, Vincent Despré, Darryl Hill, and Michiel Smid. Improved Routing on the Delaunay Triangulation. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 22:1-22:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bonichon_et_al:LIPIcs.ESA.2018.22, author = {Bonichon, Nicolas and Bose, Prosenjit and De Carufel, Jean-Lou and Despr\'{e}, Vincent and Hill, Darryl and Smid, Michiel}, title = {{Improved Routing on the Delaunay Triangulation}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {22:1--22:13}, 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.2018.22}, URN = {urn:nbn:de:0030-drops-94857}, doi = {10.4230/LIPIcs.ESA.2018.22}, annote = {Keywords: Delaunay, local routing, geometric, graph} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles.
In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira. Geodesic Obstacle Representation of Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 23:1-23:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.ICALP.2018.23, author = {Bose, Prosenjit and Carmi, Paz and Dujmovic, Vida and Mehrabi, Saeed and Montecchiani, Fabrizio and Morin, Pat and Silveira, Luis Fernando Schultz Xavier da}, title = {{Geodesic Obstacle Representation of Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.23}, URN = {urn:nbn:de:0030-drops-90274}, doi = {10.4230/LIPIcs.ICALP.2018.23}, annote = {Keywords: Obstacle representation, Grid obstacle representation, Geodesic obstacle representation} }

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

We propose faster algorithms for the following three optimization problems on n collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions.
1) Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an O(n^2)-time algorithm for this problem. We present an optimal Theta(n)-time algorithm.
2) Minimizing sum of the radii of client-server coverage: The n points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an O(n^3)-time algorithm for this problem. We present an O(n^2)-time algorithm, thereby improving the running time by a factor of Theta(n).
3) Minimizing total area of point-interval coverage: The n input points belong to an interval I. The goal is to find a set of disks of minimum total area, covering I, such that every disk contains at least one input point. We present an algorithm that solves this problem in O(n^2) time.

Ahmad Biniaz, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Ian Munro, and Michiel Smid. Faster Algorithms for some Optimization Problems on Collinear Points. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2018.8, author = {Biniaz, Ahmad and Bose, Prosenjit and Carmi, Paz and Maheshwari, Anil and Munro, Ian and Smid, Michiel}, title = {{Faster Algorithms for some Optimization Problems on Collinear Points}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {8:1--8:14}, 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.2018.8}, URN = {urn:nbn:de:0030-drops-87219}, doi = {10.4230/LIPIcs.SoCG.2018.8}, annote = {Keywords: collinear points, range assignment} }

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

Given a set of n points (sites) inside a rectangle R and n points (label locations or ports) on its boundary, a boundary labeling problem seeks ways of connecting every site to a distinct port while achieving different labeling aesthetics. We examine the scenario when the connecting lines (leaders) are drawn as axis-aligned polylines with few bends, every leader lies strictly inside R, no two leaders cross, and the sum of the lengths of all the leaders is minimized. In a k-sided boundary labeling problem, where 1 <= k <= 4, the label locations are located on the k consecutive sides of R.
In this paper we develop an O(n^3 log n)-time algorithm for 2-sided boundary labeling, where the leaders are restricted to have one bend. This improves the previously best known O(n^8 log n)-time algorithm of Kindermann et al. (Algorithmica, 76(1):225-258, 2016). We show the problem is polynomial-time solvable in more general settings such as when the ports are located on more than two sides of R, in the presence of obstacles, and even when the objective is to minimize the total number of bends. Our results improve the previous algorithms on boundary labeling with obstacles, as well as provide the first polynomial-time algorithms for minimizing the total leader length and number of bends for 3- and 4-sided boundary labeling. These results settle a number of open questions on the boundary labeling problems (Wolff, Handbook of Graph Drawing, Chapter 23, Table 23.1, 2014).

Prosenjit Bose, Paz Carmi, J. Mark Keil, Saeed Mehrabi, and Debajyoti Mondal. Boundary Labeling for Rectangular Diagrams. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2018.12, author = {Bose, Prosenjit and Carmi, Paz and Keil, J. Mark and Mehrabi, Saeed and Mondal, Debajyoti}, title = {{Boundary Labeling for Rectangular Diagrams}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {12:1--12:14}, 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.2018.12}, URN = {urn:nbn:de:0030-drops-88386}, doi = {10.4230/LIPIcs.SWAT.2018.12}, annote = {Keywords: Boundary labeling, Dynamic programming, Outerstring graphs} }

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

We consider a repulsion actuator located in an n-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an O(n^2) time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an O(n) time algorithm to find a single such actuator location if one exists. We then provide an O(n) time algorithm to place the optimal number of actuators whose sequential activation results in the gathering of the particles when such a placement exists.

Prosenjit Bose and Thomas C. Shermer. Gathering by Repulsion. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 13:1-13:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2018.13, author = {Bose, Prosenjit and Shermer, Thomas C.}, title = {{Gathering by Repulsion}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {13:1--13:12}, 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.2018.13}, URN = {urn:nbn:de:0030-drops-88397}, doi = {10.4230/LIPIcs.SWAT.2018.13}, annote = {Keywords: polygon, kernel, beacon attraction} }

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

An edge guard set of a plane graph G is a subset Gamma of edges of G such that each face of G is incident to an endpoint of an edge in Gamma. Such a set is said to guard G. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G:
1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that G can be guarded with at most 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph.
2) We prove that there exists an edge guard set of G with at most n/(3) + alpha/9 edges, where alpha is the number of quadrilateral faces in G. This improves the previous bound of n/(3) + alpha by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n/(3) edges suffice, removing the dependence on alpha.

Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, and Sander Verdonschot. Improved Bounds for Guarding Plane Graphs with Edges. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 14:1-14:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biniaz_et_al:LIPIcs.SWAT.2018.14, author = {Biniaz, Ahmad and Bose, Prosenjit and Ooms, Aur\'{e}lien and Verdonschot, Sander}, title = {{Improved Bounds for Guarding Plane Graphs with Edges}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {14:1--14:12}, 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.2018.14}, URN = {urn:nbn:de:0030-drops-88403}, doi = {10.4230/LIPIcs.SWAT.2018.14}, annote = {Keywords: edge guards, graph coloring, four-color theorem} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let P be a set of n vertices in the plane and let S be a set of line segments between the vertices in P, with no two line segments intersecting properly. We present two 1-local O(1)-memory routing algorithms on the visibility graph of P with respect to a set of constraints S (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of information). Contrary to all existing routing algorithms, our routing algorithms do not require us to compute a plane subgraph of the visibility graph in order to route on it.

Prosenjit Bose, Matias Korman, André van Renssen, and Sander Verdonschot. Routing on the Visibility Graph. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 18:1-18:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bose_et_al:LIPIcs.ISAAC.2017.18, author = {Bose, Prosenjit and Korman, Matias and van Renssen, Andr\'{e} and Verdonschot, Sander}, title = {{Routing on the Visibility Graph}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {18:1--18:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.18}, URN = {urn:nbn:de:0030-drops-82224}, doi = {10.4230/LIPIcs.ISAAC.2017.18}, annote = {Keywords: Routing, constraints, visibility graph, Theta-graph} }

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

We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| >= |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order.
Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.

Prosenjit Bose, Irina Kostitsyna, and Stefan Langerman. Self-Approaching Paths in Simple Polygons. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2017.21, author = {Bose, Prosenjit and Kostitsyna, Irina and Langerman, Stefan}, title = {{Self-Approaching Paths in Simple Polygons}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {21:1--21:15}, 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.2017.21}, URN = {urn:nbn:de:0030-drops-72166}, doi = {10.4230/LIPIcs.SoCG.2017.21}, annote = {Keywords: self-approaching path, simple polygon, shortest path, involute curve} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Let S be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane frac{3+4 pi}{3}-spanner of S whose vertex degree is at most 3. Let Lambda be the vertex set of a finite non-uniform rectangular lattice in the plane. We present an algorithm that constructs a plane 3 sqrt{2}-spanner for Lambda whose vertex degree is at most 3. For points that are in the plane and in general position, we show how to compute plane degree-3 spanners with a linear number of Steiner points.

Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, Anil Maheshwari, and Michiel Smid. Towards Plane Spanners of Degree 3. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 19:1-19:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{biniaz_et_al:LIPIcs.ISAAC.2016.19, author = {Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Gavoille, Cyril and Maheshwari, Anil and Smid, Michiel}, title = {{Towards Plane Spanners of Degree 3}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.19}, URN = {urn:nbn:de:0030-drops-67887}, doi = {10.4230/LIPIcs.ISAAC.2016.19}, annote = {Keywords: plane spanners, degree-3 spanners, convex position, non-uniform lattice} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).

Mahdi Amani, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Anil Maheshwari, and Michiel Smid. A Plane 1.88-Spanner for Points in Convex Position. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 25:1-25:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{amani_et_al:LIPIcs.SWAT.2016.25, author = {Amani, Mahdi and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Maheshwari, Anil and Smid, Michiel}, title = {{A Plane 1.88-Spanner for Points in Convex Position}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.25}, URN = {urn:nbn:de:0030-drops-60474}, doi = {10.4230/LIPIcs.SWAT.2016.25}, annote = {Keywords: points in convex position, plane spanner} }

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

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

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

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@InProceedings{ahn_et_al:LIPIcs.SOCG.2015.209, author = {Ahn, Hee Kap and Barba, Luis and Bose, Prosenjit and De Carufel, Jean-Lou and Korman, Matias and Oh, Eunjin}, title = {{A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {209--223}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.209}, URN = {urn:nbn:de:0030-drops-51448}, doi = {10.4230/LIPIcs.SOCG.2015.209}, annote = {Keywords: Geodesic distance, facility location, 1-center problem, simple polygons} }

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