11 Search Results for "Polishchuk, Valentin"


Document
On Flipping the Fréchet Distance

Authors: Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min - the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Cite as

Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk. On Flipping the Fréchet Distance. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.51,
  author =	{Filtser, Omrit and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin},
  title =	{{On Flipping the Fr\'{e}chet Distance}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{51:1--51:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.51},
  URN =		{urn:nbn:de:0030-drops-175548},
  doi =		{10.4230/LIPIcs.ITCS.2023.51},
  annote =	{Keywords: curves, polygons, distancing measure}
}
Document
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Authors: Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth

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


Abstract
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.

Cite as

Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{arkin_et_al:LIPIcs.ESA.2020.7,
  author =	{Arkin, Esther M. and Das, Rathish and Gao, Jie and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin and T\'{o}th, Csaba D.},
  title =	{{Cutting Polygons into Small Pieces with Chords: Laser-Based Localization}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.7},
  URN =		{urn:nbn:de:0030-drops-128736},
  doi =		{10.4230/LIPIcs.ESA.2020.7},
  annote =	{Keywords: Polygon partition, Arrangements, Visibility, Localization}
}
Document
Geometric Secluded Paths and Planar Satisfiability

Authors: Kevin Buchin, Valentin Polishchuk, Leonid Sedov, and Roman Voronov

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


Abstract
We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT.

Cite as

Kevin Buchin, Valentin Polishchuk, Leonid Sedov, and Roman Voronov. Geometric Secluded Paths and Planar Satisfiability. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{buchin_et_al:LIPIcs.SoCG.2020.24,
  author =	{Buchin, Kevin and Polishchuk, Valentin and Sedov, Leonid and Voronov, Roman},
  title =	{{Geometric Secluded Paths and Planar Satisfiability}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.24},
  URN =		{urn:nbn:de:0030-drops-121827},
  doi =		{10.4230/LIPIcs.SoCG.2020.24},
  annote =	{Keywords: Visibility, Route planning, Security/privacy, Planar satisfiability}
}
Document
New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs

Authors: Nil Mamano, Alon Efrat, David Eppstein, Daniel Frishberg, Michael T. Goodrich, Stephen Kobourov, Pedro Matias, and Valentin Polishchuk

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


Abstract
We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0.

Cite as

Nil Mamano, Alon Efrat, David Eppstein, Daniel Frishberg, Michael T. Goodrich, Stephen Kobourov, Pedro Matias, and Valentin Polishchuk. New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 51:1-51:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{mamano_et_al:LIPIcs.ISAAC.2019.51,
  author =	{Mamano, Nil and Efrat, Alon and Eppstein, David and Frishberg, Daniel and Goodrich, Michael T. and Kobourov, Stephen and Matias, Pedro and Polishchuk, Valentin},
  title =	{{New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{51:1--51:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.51},
  URN =		{urn:nbn:de:0030-drops-115477},
  doi =		{10.4230/LIPIcs.ISAAC.2019.51},
  annote =	{Keywords: Nearest-neighbors, Nearest-neighbor chain, motorcycle graph, straight skeleton, multi-fragment algorithm, Euclidean TSP, Steiner TSP}
}
Document
Triangles and Girth in Disk Graphs and Transmission Graphs

Authors: Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

Cite as

Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Triangles and Girth in Disk Graphs and Transmission Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{kaplan_et_al:LIPIcs.ESA.2019.64,
  author =	{Kaplan, Haim and Klost, Katharina and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha},
  title =	{{Triangles and Girth in Disk Graphs and Transmission Graphs}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{64:1--64:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.64},
  URN =		{urn:nbn:de:0030-drops-111859},
  doi =		{10.4230/LIPIcs.ESA.2019.64},
  annote =	{Keywords: disk graph, transmission graph, triangle, girth}
}
Document
Gender-Aware Facility Location in Multi-Gender World

Authors: Valentin Polishchuk and Leonid Sedov

Published in: LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)


Abstract
This interdisciplinary (GS and CS) paper starts from considering the problem of locating restrooms or locker rooms in a privacy-preserving way, i.e., so that while following the path to one's room, one cannot peek into another room; the rooms are meant for a multitude of genders, one room per gender. We then proceed to showing that gender inequality (non-uniform treatment of genders by genders) makes the room placement hard. Finally, we delve into specifics of gender definition and consider locating facilities for the genders in a "perfect" way, i.e., so that navigating to the facilities involves only quick binary decisions; on the way, we indicate that there is room for interpretation the facilities under consideration (we outline several possibilities, depending on the application).

Cite as

Valentin Polishchuk and Leonid Sedov. Gender-Aware Facility Location in Multi-Gender World. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{polishchuk_et_al:LIPIcs.FUN.2018.28,
  author =	{Polishchuk, Valentin and Sedov, Leonid},
  title =	{{Gender-Aware Facility Location in Multi-Gender World}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.28},
  URN =		{urn:nbn:de:0030-drops-88191},
  doi =		{10.4230/LIPIcs.FUN.2018.28},
  annote =	{Keywords: visibility, Strahler number, perfect tree, interval graphs, gender studies}
}
Document
Automatic Design of Aircraft Arrival Routes with Limited Turning Angle

Authors: Tobias Andersson Granberg, Tatiana Polishchuk, Valentin Polishchuk, and Christiane Schmidt

Published in: OASIcs, Volume 54, 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)


Abstract
We present an application of Integer Programming to the design of arrival routes for aircraft in a Terminal Maneuvering Area (TMA). We generate operationally feasible merge trees of curvature-constrained routes, using two optimization criteria: (1) total length of the tree, and (2) distance flown along the tree paths. The output routes guarantee that the overall traffic pattern in the TMA can be monitored by air traffic controllers; in particular, we keep merge points for arriving aircraft well separated, and we exclude conflicts between arriving and departing aircraft. We demonstrate the feasibility of our method by experimenting with arrival routes for a runway at Arlanda airport in the Stockholm TMA. Our approach can easily be extended in several ways, e.g., to ensure that the routes avoid no-fly zones.

Cite as

Tobias Andersson Granberg, Tatiana Polishchuk, Valentin Polishchuk, and Christiane Schmidt. Automatic Design of Aircraft Arrival Routes with Limited Turning Angle. In 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016). Open Access Series in Informatics (OASIcs), Volume 54, pp. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{granberg_et_al:OASIcs.ATMOS.2016.9,
  author =	{Granberg, Tobias Andersson and Polishchuk, Tatiana and Polishchuk, Valentin and Schmidt, Christiane},
  title =	{{Automatic Design of Aircraft Arrival Routes with Limited Turning Angle}},
  booktitle =	{16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)},
  pages =	{9:1--9:13},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-021-7},
  ISSN =	{2190-6807},
  year =	{2016},
  volume =	{54},
  editor =	{Goerigk, Marc and Werneck, Renato F.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2016.9},
  URN =		{urn:nbn:de:0030-drops-65336},
  doi =		{10.4230/OASIcs.ATMOS.2016.9},
  annote =	{Keywords: Air Traffic Management, Standard Terminal Arrival Routes, Standard Instrument Departures, Integer programming, Turn constraints}
}
Document
On the Complexity of Minimum-Link Path Problems

Authors: Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk, and Frank Staals

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


Abstract
We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the min-link path's vertices or edges can be restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several versions of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al. 2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al. 1992] mentioned in the handbook [Goodman and O'Rourke, 2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [Demaine et al. TOPP] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

Cite as

Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk, and Frank Staals. On the Complexity of Minimum-Link Path Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{kostitsyna_et_al:LIPIcs.SoCG.2016.49,
  author =	{Kostitsyna, Irina and L\"{o}ffler, Maarten and Polishchuk, Valentin and Staals, Frank},
  title =	{{On the Complexity of Minimum-Link Path Problems}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{49:1--49:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.49},
  URN =		{urn:nbn:de:0030-drops-59412},
  doi =		{10.4230/LIPIcs.SoCG.2016.49},
  annote =	{Keywords: minimum-linkpath, diffuse reflection, terrain, bit complexity, NP-hardness}
}
Document
Recognizing a DOG is Hard, But Not When It is Thin and Unit

Authors: William Evans, Mereke van Garderen, Maarten Löffler, and Valentin Polishchuk

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)


Abstract
We define the notion of disk-obedience for a set of disks in the plane and give results for diskobedient graphs (DOGs), which are disk intersection graphs (DIGs) that admit a planar embedding with vertices inside the corresponding disks. We show that in general it is hard to recognize a DOG, but when the DIG is thin and unit (i.e., when the disks are unit disks), it can be done in linear time.

Cite as

William Evans, Mereke van Garderen, Maarten Löffler, and Valentin Polishchuk. Recognizing a DOG is Hard, But Not When It is Thin and Unit. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{evans_et_al:LIPIcs.FUN.2016.16,
  author =	{Evans, William and van Garderen, Mereke and L\"{o}ffler, Maarten and Polishchuk, Valentin},
  title =	{{Recognizing a DOG is Hard, But Not When It is Thin and Unit}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{16:1--16:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.16},
  URN =		{urn:nbn:de:0030-drops-58671},
  doi =		{10.4230/LIPIcs.FUN.2016.16},
  annote =	{Keywords: graph drawing, planar graphs, disk intersection graphs}
}
Document
Computing the L1 Geodesic Diameter and Center of a Polygonal Domain

Authors: Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, and Haitao Wang

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)


Abstract
For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the L_1 geodesic diameter in O(n^2+h^4) time and the L_1 geodesic center in O((n^4+n^2 h^4)*alpha(n)) time, where alpha(.) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in O(n^{7.73}) or O(n^7(h+log(n))) time, and compute the geodesic center in O(n^{12+epsilon}) time. Therefore, our algorithms are much faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on L_1 shortest paths in polygonal domains.

Cite as

Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, and Haitao Wang. Computing the L1 Geodesic Diameter and Center of a Polygonal Domain. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{wonbae_et_al:LIPIcs.STACS.2016.14,
  author =	{Won Bae, Sang and Korman, Matias and Mitchell, Joseph S. B. and Okamoto, Yoshio and Polishchuk, Valentin and Wang, Haitao},
  title =	{{Computing the L1 Geodesic Diameter and Center of a Polygonal Domain}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.14},
  URN =		{urn:nbn:de:0030-drops-57151},
  doi =		{10.4230/LIPIcs.STACS.2016.14},
  annote =	{Keywords: geodesic diameter, geodesic center, shortest paths, polygonal domains, L1 metric}
}
Document
Shortest Path to a Segment and Quickest Visibility Queries

Authors: Esther M. Arkin, Alon Efrat, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, and Topi Talvitie

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


Abstract
We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.

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Esther M. Arkin, Alon Efrat, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, and Topi Talvitie. Shortest Path to a Segment and Quickest Visibility Queries. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 658-673, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{arkin_et_al:LIPIcs.SOCG.2015.658,
  author =	{Arkin, Esther M. and Efrat, Alon and Knauer, Christian and Mitchell, Joseph S. B. and Polishchuk, Valentin and Rote, G\"{u}nter and Schlipf, Lena and Talvitie, Topi},
  title =	{{Shortest Path to a Segment and Quickest Visibility Queries}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{658--673},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.658},
  URN =		{urn:nbn:de:0030-drops-51474},
  doi =		{10.4230/LIPIcs.SOCG.2015.658},
  annote =	{Keywords: path planning, visibility, query structures and complexity, persistent data structures, continuous Dijkstra}
}
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