7 Search Results for "Kobourov, Stephen G."


Document
Multi-Level Weighted Additive Spanners

Authors: Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)


Abstract
Given a graph G = (V,E), a subgraph H is an additive +β spanner if dist_H(u,v) ≤ dist_G(u,v) + β for all u, v ∈ V. A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V, it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several existing algorithms by [Ahmed et al., 2020] for weighted additive spanners, both in terms of runtime and sparsity of the output spanner, when applied as a subroutine to multi-level problem. We provide an experimental evaluation on graphs using several different random graph generators and show that these spanner algorithms typically achieve much better guarantees in terms of sparsity and additive error compared with the theoretical maximum. By analyzing our experimental results, we additionally developed a new technique of changing a certain initialization parameter which provides better spanners in practice at the expense of a small increase in running time.

Cite as

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Multi-Level Weighted Additive Spanners. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{ahmed_et_al:LIPIcs.SEA.2021.16,
  author =	{Ahmed, Reyan and Bodwin, Greg and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
  title =	{{Multi-Level Weighted Additive Spanners}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{16:1--16:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.16},
  URN =		{urn:nbn:de:0030-drops-137885},
  doi =		{10.4230/LIPIcs.SEA.2021.16},
  annote =	{Keywords: multi-level, graph spanner, approximation algorithms}
}
Document
Computing β-Stretch Paths in Drawings of Graphs

Authors: Esther M. Arkin, Faryad Darabi Sahneh, Alon Efrat, Fabian Frank, Radoslav Fulek, Stephen Kobourov, and Joseph S. B. Mitchell

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


Abstract
Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π=f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p∈P and q∈P, the length of the sub-curve of π connecting f(p) with f(q) is at most β||f(p)-f(q)‖, where ‖.‖ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. The βSP also asks that we output P if it exists. The βSP quantifies a notion of "near straightness" for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε>0, β∈O(poly(log |V(G)|)), and s,t∈V(G), outputs a β-stretch path between s and t, if a (1-ε)β-stretch path between s and t exists in the drawing.

Cite as

Esther M. Arkin, Faryad Darabi Sahneh, Alon Efrat, Fabian Frank, Radoslav Fulek, Stephen Kobourov, and Joseph S. B. Mitchell. Computing β-Stretch Paths in Drawings of Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{arkin_et_al:LIPIcs.SWAT.2020.7,
  author =	{Arkin, Esther M. and Sahneh, Faryad Darabi and Efrat, Alon and Frank, Fabian and Fulek, Radoslav and Kobourov, Stephen and Mitchell, Joseph S. B.},
  title =	{{Computing \beta-Stretch Paths in Drawings of Graphs}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{7:1--7:20},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.7},
  URN =		{urn:nbn:de:0030-drops-122540},
  doi =		{10.4230/LIPIcs.SWAT.2020.7},
  annote =	{Keywords: stretch factor, dilation, geometric spanners}
}
Document
Multi-Level Steiner Trees

Authors: Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G. Kobourov, Richard Spence, Joseph Watkins, and Alexander Wolff

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)


Abstract
In the classical Steiner tree problem, one is given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T subseteq V. The objective is to find a minimum-cost edge set E' subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of rho = ln(4)+epsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 subset ... subset T_k subseteq V, compute nested edge sets E_1 subseteq ... subseteq E_k subseteq E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth. Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k <= 22), which works well for applications such as designing multi-level infrastructure or network visualization.

Cite as

Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G. Kobourov, Richard Spence, Joseph Watkins, and Alexander Wolff. Multi-Level Steiner Trees. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{ahmed_et_al:LIPIcs.SEA.2018.15,
  author =	{Ahmed, Reyan and Angelini, Patrizio and Sahneh, Faryad Darabi and Efrat, Alon and Glickenstein, David and Gronemann, Martin and Heinsohn, Niklas and Kobourov, Stephen G. and Spence, Richard and Watkins, Joseph and Wolff, Alexander},
  title =	{{Multi-Level Steiner Trees}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.15},
  URN =		{urn:nbn:de:0030-drops-89506},
  doi =		{10.4230/LIPIcs.SEA.2018.15},
  annote =	{Keywords: Approximation algorithm, Steiner tree, multi-level graph representation}
}
Document
Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

Authors: Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, and János Pach

Published in: Dagstuhl Reports, Volume 6, Issue 11 (2017)


Abstract
This report summarizes Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics'' and documents the talks and discussions. The seminar brought together 29 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing. The common interest was in the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs with topological constraints such as specific types of crossings, or with some forbidden crossing patterns. The seminar began with three introductory talks by experts in the different fields. Abstracts of these talks are collected in this report. Next we discussed and grouped together open research problems about beyond planar graphs, such as their combinatorial structures (e.g, thickness, crossing number, coloring), their topology (e.g., string graph representation), their geometric representations (e.g., straight-line drawing, visibility representation, contact representation), and applications (e.g., algorithms for real-world network visualization). Four working groups were formed and a report from each group is included here.

Cite as

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, and János Pach. Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452). In Dagstuhl Reports, Volume 6, Issue 11, pp. 35-62, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@Article{hong_et_al:DagRep.6.11.35,
  author =	{Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. and Pach, J\'{a}nos},
  title =	{{Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)}},
  pages =	{35--62},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{6},
  number =	{11},
  editor =	{Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. 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/DagRep.6.11.35},
  URN =		{urn:nbn:de:0030-drops-70385},
  doi =		{10.4230/DagRep.6.11.35},
  annote =	{Keywords: graph drawing, graph algorithms, graph theory, geometric algorithms, combinatorial geometry, visualization}
}
Document
Drawing Graphs and Maps with Curves (Dagstuhl Seminar 13151)

Authors: Stephen G. Kobourov, Martin Nöllenburg, and Monique Teillaud

Published in: Dagstuhl Reports, Volume 3, Issue 4 (2013)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 13151 "Drawing Graphs and Maps with Curves". The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants.

Cite as

Stephen G. Kobourov, Martin Nöllenburg, and Monique Teillaud. Drawing Graphs and Maps with Curves (Dagstuhl Seminar 13151). In Dagstuhl Reports, Volume 3, Issue 4, pp. 34-68, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@Article{kobourov_et_al:DagRep.3.4.34,
  author =	{Kobourov, Stephen G. and N\"{o}llenburg, Martin and Teillaud, Monique},
  title =	{{Drawing Graphs and Maps with Curves (Dagstuhl Seminar 13151)}},
  pages =	{34--68},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{3},
  number =	{4},
  editor =	{Kobourov, Stephen G. and N\"{o}llenburg, Martin and Teillaud, Monique},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.3.4.34},
  URN =		{urn:nbn:de:0030-drops-41680},
  doi =		{10.4230/DagRep.3.4.34},
  annote =	{Keywords: graph drawing, information visualization, computational cartography, computational geometry}
}
Document
08191 Working Group Report – X-graphs of Y-graphs and their Representations

Authors: Vladimir Batagelj, Franz J. Brandenburg, Walter Didimo, Guiseppe Liotta, and Maurizio Patrignani

Published in: Dagstuhl Seminar Proceedings, Volume 8191, Graph Drawing with Applications to Bioinformatics and Social Sciences (2008)


Abstract
We address graph decomposition problems that help the hybrid visualization of large graphs, where different graphic metaphors (node-link, matrix, etc.) are used in the same picture. We generalize the $X$-graphs of $Y$-graphs model introduced by Brandenburg (Brandenburg, F.J.: Graph clustering I: Cycles of cliques. In Di Battista, G., ed.: Graph Drawing (Proc. GD '97). Volume 1353 of Lecture Notes Comput. Sci., Springer-Verlag (1997) 158--168) to formalize the problem of automatically identifying dense subgraphs ($Y$-graphs, clusters) that are prone to be collapsed and shown with a matricial representation when needed. We show that (planar, $K_5$)-recognition, that is, the problem of identifying $K_5$ subgraphs such that the graph obtained by collapsing them is planar, is NP-hard. On the positive side, we show that it is possible to determine the highest value of $k$ such that $G$ is a (planar,$k$-core)-graph in $O(m + n log(n))$ time.

Cite as

Vladimir Batagelj, Franz J. Brandenburg, Walter Didimo, Guiseppe Liotta, and Maurizio Patrignani. 08191 Working Group Report – X-graphs of Y-graphs and their Representations. In Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl Seminar Proceedings, Volume 8191, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{batagelj_et_al:DagSemProc.08191.5,
  author =	{Batagelj, Vladimir and Brandenburg, Franz J. and Didimo, Walter and Liotta, Guiseppe and Patrignani, Maurizio},
  title =	{{08191 Working Group Report – X-graphs of Y-graphs and their Representations}},
  booktitle =	{Graph Drawing with Applications to Bioinformatics and Social Sciences},
  pages =	{1--17},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8191},
  editor =	{Stephen P. Borgatti and Stephen Kobourov and Oliver Kohlbacher and Petra Mutzel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08191.5},
  URN =		{urn:nbn:de:0030-drops-15563},
  doi =		{10.4230/DagSemProc.08191.5},
  annote =	{Keywords: Graph drawing, X-graphs of Y-graphs, visualization of large graphs}
}
Document
Force-Directed Approaches to Sensor Network Localization

Authors: Stephen G. Kobourov, Alon Efrat, David Forrester, and Anand Iyer

Published in: Dagstuhl Seminar Proceedings, Volume 5361, Algorithmic Aspects of Large and Complex Networks (2006)


Abstract
In many sensor network applications it is necessary to compute low-error localization of the sensor nodes. Although embedding a GPS unit on each node would solve the problem for many outdoor applications, the cost of this solution for large networks is prohibitively high. We consider static and mobile network localization approaches that make use of the local neighborhood information, in the form of relative distances and angles to nearby nodes, gathered through simpler and less costly devices (RF, ultrasound based range sensors, or antenna arrays). Our algorithms do not make any assumptions about the existence of anchor nodes capable of locating themselves, nor about the knowledge of an initial localization to start with. Instead, we rely on a multi-scale force-directed approach, utilizing range and angle data through dead reckoning. We show that our localization algorithms are robust and scale well with network size.

Cite as

Stephen G. Kobourov, Alon Efrat, David Forrester, and Anand Iyer. Force-Directed Approaches to Sensor Network Localization. In Algorithmic Aspects of Large and Complex Networks. Dagstuhl Seminar Proceedings, Volume 5361, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)


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@InProceedings{kobourov_et_al:DagSemProc.05361.7,
  author =	{Kobourov, Stephen G. and Efrat, Alon and Forrester, David and Iyer, Anand},
  title =	{{Force-Directed Approaches to Sensor Network Localization}},
  booktitle =	{Algorithmic Aspects of Large and Complex Networks},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5361},
  editor =	{Stefano Leonardi and Friedhelm Meyer auf der Heide and Dorothea Wagner},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.05361.7},
  URN =		{urn:nbn:de:0030-drops-5693},
  doi =		{10.4230/DagSemProc.05361.7},
  annote =	{Keywords: Sensor network localization, multi-scale force-directed approach, dead reckoning}
}
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