Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

In this work, we address the following question. Suppose we are given a set D of positive-weighted disks and a set T of n points in the plane, such that each point of T is contained in at least two disks of D. Then is there always a subset S of D such that the union of the disks in S contains all the points of T and the total weight of the disks of D that are not in S is at least a constant fraction of the total weight of the disks in D?
In our work, we prove the Extraction Theorem that answers this question in the affirmative. Our constructive proof heavily exploits the geometry of disks, and in the process, we make interesting connections between our work and the literature on local search for geometric optimization problems.
The Extraction Theorem helps to design the first polynomial-time O(1)-approximations for two important geometric covering problems involving disks.

Sayan Bandyapadhyay, Anil Maheshwari, Sasanka Roy, Michiel Smid, and Kasturi Varadarajan. Geometric Covering via Extraction Theorem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ITCS.2024.7, author = {Bandyapadhyay, Sayan and Maheshwari, Anil and Roy, Sasanka and Smid, Michiel and Varadarajan, Kasturi}, title = {{Geometric Covering via Extraction Theorem}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.7}, URN = {urn:nbn:de:0030-drops-195355}, doi = {10.4230/LIPIcs.ITCS.2024.7}, annote = {Keywords: Covering, Extraction theorem, Double-disks, Submodularity, Local search} }

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

Given two sets S and T of points in the plane, of total size n, a many-to-many matching between S and T is a set of pairs (p,q) such that p ∈ S, q ∈ T and for each r ∈ S ∪ T, r appears in at least one such pair. The cost of a pair (p,q) is the (Euclidean) distance between p and q. In the minimum-cost many-to-many matching problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in O(n³) time. In a more restricted setting where all the points are on a line, the problem can be solved in O(nlog n) time [Justin Colannino et al., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an O(n²⋅ poly(log n)) time exact algorithm and an O(n^{3/2}⋅ poly(log n)) time (1+ε)-approximation in the planar case.

Sayan Bandyapadhyay, Anil Maheshwari, and Michiel Smid. Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ISAAC.2021.44, author = {Bandyapadhyay, Sayan and Maheshwari, Anil and Smid, Michiel}, title = {{Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {44:1--44:14}, 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.2021.44}, URN = {urn:nbn:de:0030-drops-154779}, doi = {10.4230/LIPIcs.ISAAC.2021.44}, annote = {Keywords: Many-to-many matching, bipartite, planar, geometric, approximation} }

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

A beer graph is an undirected graph G, in which each edge has a positive weight and some vertices have a beer store. A beer path between two vertices u and v in G is any path in G between u and v that visits at least one beer store.
We show that any outerplanar beer graph G with n vertices can be preprocessed in O(n) time into a data structure of size O(n), such that for any two query vertices u and v, (i) the weight of the shortest beer path between u and v can be reported in O(α(n)) time (where α(n) is the inverse Ackermann function), and (ii) the shortest beer path between u and v can be reported in O(L) time, where L is the number of vertices on this path. Both results are optimal, even when G is a beer tree (i.e., a beer graph whose underlying graph is a tree).

Joyce Bacic, Saeed Mehrabi, and Michiel Smid. Shortest Beer Path Queries in Outerplanar Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bacic_et_al:LIPIcs.ISAAC.2021.62, author = {Bacic, Joyce and Mehrabi, Saeed and Smid, Michiel}, title = {{Shortest Beer Path Queries in Outerplanar Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {62:1--62:16}, 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.2021.62}, URN = {urn:nbn:de:0030-drops-154950}, doi = {10.4230/LIPIcs.ISAAC.2021.62}, annote = {Keywords: shortest paths, outerplanar graph} }

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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 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We consider a well studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and an integer d >= 1, in the maximum diameter-bounded subgraph problem (MaxDBS for short), the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d=1, this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n^{1-epsilon}, for any epsilon > 0. Moreover, it is known that, for any d >= 2, it is NP-hard to approximate MaxDBS within a factor n^{1/2 - epsilon}, for any epsilon > 0.
In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs.

A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, and Shakhar Smorodinsky. Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abuaffash_et_al:LIPIcs.SoCG.2018.2, author = {Abu-Affash, A. Karim and Carmi, Paz and Maheshwari, Anil and Morin, Pat and Smid, Michiel and Smorodinsky, Shakhar}, title = {{Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {2:1--2:12}, 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.2018.2}, URN = {urn:nbn:de:0030-drops-87152}, doi = {10.4230/LIPIcs.SoCG.2018.2}, annote = {Keywords: Approximation algorithms, maximum diameter-bounded subgraph, unit disk graphs, fractional Helly theorem, VC-dimension} }

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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} }

Document

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

We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n^2) and by van Leeuwen and Schoone (1980) that F(M)=O(n^3). We prove that f(M)=O(n Delta) where Delta is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2-1 and F(M)={{n/2} choose 2}; these bounds are tight.
Any bound on F(*) carries over to the bichromatic setting, while this is not necessarily true for f(*). Let M' be a bichromatic matching. The best known upper bound for f(M') is the same as for F(M'), which is essentially O(n^3). We prove that f(M')<=slant n-2 for points in convex position, and f(M')= O(n^2) for semi-collinear points.
The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(n log n).

Ahmad Biniaz, Anil Maheshwari, and Michiel Smid. Flip Distance to some Plane Configurations. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biniaz_et_al:LIPIcs.SWAT.2018.11, author = {Biniaz, Ahmad and Maheshwari, Anil and Smid, Michiel}, title = {{Flip Distance to some Plane Configurations}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {11:1--11:14}, 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.2018.11}, URN = {urn:nbn:de:0030-drops-88371}, doi = {10.4230/LIPIcs.SWAT.2018.11}, annote = {Keywords: flip distance, non-crossing edges, perfect matchings, spanning trees} }

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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 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network.
We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves.
Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices.

Jean-Lou De Carufel, Carsten Grimm, Anil Maheshwari, and Michiel Smid. Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{decarufel_et_al:LIPIcs.SWAT.2016.27, author = {De Carufel, Jean-Lou and Grimm, Carsten and Maheshwari, Anil and Smid, Michiel}, title = {{Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {27:1--27:14}, 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.2016.27}, URN = {urn:nbn:de:0030-drops-60492}, doi = {10.4230/LIPIcs.SWAT.2016.27}, annote = {Keywords: Network Augmentation, Shortcuts, Diameter, Paths, Cycles} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9451, Geometric Networks, Metric Space Embeddings and Spatial Data Mining (2010)

From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining'' was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Gautam Das, Joachim Gudmundsson, Rolf Klein, Christian Knauer, and Michiel Smid. 09451 Abstracts Collection – Geometric Networks, Metric Space Embeddings and Spatial Data Mining. In Geometric Networks, Metric Space Embeddings and Spatial Data Mining. Dagstuhl Seminar Proceedings, Volume 9451, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{das_et_al:DagSemProc.09451.1, author = {Das, Gautam and Gudmundsson, Joachim and Klein, Rolf and Knauer, Christian and Smid, Michiel}, title = {{09451 Abstracts Collection – Geometric Networks, Metric Space Embeddings and Spatial Data Mining}}, booktitle = {Geometric Networks, Metric Space Embeddings and Spatial Data Mining}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9451}, editor = {Gautam Das and Joachim Gudmundsson and Rolf Klein and Christian Knauer and Michiel Smid}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09451.1}, URN = {urn:nbn:de:0030-drops-24380}, doi = {10.4230/DagSemProc.09451.1}, annote = {Keywords: Geometric networks, metric space embeddings, spatial data mining, spanners, dilation, distortion} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6481, Geometric Networks and Metric Space Embeddings (2007)

The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space
Embeddings'' was held from November~26 to December~1, 2006 in the
International Conference and Research Center (IBFI), Schloss
Dagstuhl. During the seminar, several participants presented their
current research, and ongoing work and open problems were discussed.
In this paper we describe the seminar topics, we have compiled a
list of open questions that were posed during the seminar, there is
a list of all talks and there are abstracts of the presentations
given during the seminar. Links to extended abstracts or full
papers are provided where available.

Joachim Gudmundsson, Rolf Klein, Giri Narasimhan, Michiel Smid, and Alexander Wolff. 06481 Abstracts Collection – Geometric Networks and Metric Space Embeddings. In Geometric Networks and Metric Space Embeddings. Dagstuhl Seminar Proceedings, Volume 6481, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{gudmundsson_et_al:DagSemProc.06481.1, author = {Gudmundsson, Joachim and Klein, Rolf and Narasimhan, Giri and Smid, Michiel and Wolff, Alexander}, title = {{06481 Abstracts Collection – Geometric Networks and Metric Space Embeddings}}, booktitle = {Geometric Networks and Metric Space Embeddings}, pages = {1--21}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {6481}, editor = {Joachim Gudmundsson and Rolf Klein and Giri Narasimhan and Michiel Smid and Alexander Wolff}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06481.1}, URN = {urn:nbn:de:0030-drops-10291}, doi = {10.4230/DagSemProc.06481.1}, annote = {Keywords: Geometric networks, metric space embeddings, phylogenetic networks, spanners, dilation, distortion} }