5 Search Results for "Knauer, Christian"


Document
On the Stretch Factor of Polygonal Chains

Authors: Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.

Cite as

Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{chen_et_al:LIPIcs.MFCS.2019.56,
  author =	{Chen, Ke and Dumitrescu, Adrian and Mulzer, Wolfgang and T\'{o}th, Csaba D.},
  title =	{{On the Stretch Factor of Polygonal Chains}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.56},
  URN =		{urn:nbn:de:0030-drops-110005},
  doi =		{10.4230/LIPIcs.MFCS.2019.56},
  annote =	{Keywords: polygonal chain, vertex dilation, Koch curve, recursive construction}
}
Document
Placing your Coins on a Shelf

Authors: Helmut Alt, Kevin Buchin, Steven Chaplick, Otfried Cheong, Philipp Kindermann, Christian Knauer, and Fabian Stehn

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


Abstract
We consider the problem of packing a family of disks 'on a shelf,' that is, such that each disk touches the x-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in O(n log n) time, and provide an O(n log n)-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.

Cite as

Helmut Alt, Kevin Buchin, Steven Chaplick, Otfried Cheong, Philipp Kindermann, Christian Knauer, and Fabian Stehn. Placing your Coins on a Shelf. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{alt_et_al:LIPIcs.ISAAC.2017.4,
  author =	{Alt, Helmut and Buchin, Kevin and Chaplick, Steven and Cheong, Otfried and Kindermann, Philipp and Knauer, Christian and Stehn, Fabian},
  title =	{{Placing your Coins on a Shelf}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{4:1--4:12},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.4},
  URN =		{urn:nbn:de:0030-drops-82145},
  doi =		{10.4230/LIPIcs.ISAAC.2017.4},
  annote =	{Keywords: packing problems, approximation algorithms, NP-hardness}
}
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.

Cite as

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}
}
Document
On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems

Authors: Christian Knauer, Hans Raj Tiwary, and Daniel Werner

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)


Abstract
We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the hs cut problem are W[1]-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Our reductions also imply that the problems we consider cannot be solved in time n^{o(d)} (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. The technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.

Cite as

Christian Knauer, Hans Raj Tiwary, and Daniel Werner. On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 649-660, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{knauer_et_al:LIPIcs.STACS.2011.649,
  author =	{Knauer, Christian and Tiwary, Hans Raj and Werner, Daniel},
  title =	{{On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{649--660},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.649},
  URN =		{urn:nbn:de:0030-drops-30514},
  doi =		{10.4230/LIPIcs.STACS.2011.649},
  annote =	{Keywords: computational geometry, combinatorial geometry, ham-sandwich cuts, parameterized complexity.}
}
Document
09451 Abstracts Collection – Geometric Networks, Metric Space Embeddings and Spatial Data Mining

Authors: Gautam Das, Joachim Gudmundsson, Rolf Klein, Christian Knauer, and Michiel Smid

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


Abstract
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.

Cite as

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-dev.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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