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Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs [article]

Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, André Nusser, Zahra Parsaeian
2022 arXiv   pre-print
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective.  ...  Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since  ...  In special graph classes however it is possible to compute the diameter in sub-quadratic time.  ... 
arXiv:2203.03663v2 fatcat:xi4otutaivarhokb23qhaw66qa

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, Zahra Parsaeian, Xavier Goaoc, Michael Kerber
2022
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective.  ...  Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since  ...  In special graph classes however it is possible to compute the diameter in sub-quadratic time.  ... 
doi:10.4230/lipics.socg.2022.21 fatcat:ncvi4sxorzd63ist6l6jo4ol34

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, Zahra Parsaeian, Xavier Goaoc, Michael Kerber
2022
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective.  ...  Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since  ...  In special graph classes however it is possible to compute the diameter in sub-quadratic time.  ... 
doi:10.3929/ethz-b-000553719 fatcat:b5kkm74gjnhehfajqzdjjcfqxa

EUCLIDEAN CONSTRUCTION FOR IMAGINARY ROOTS OF THE QUADRATIC EQUATION

J. Shaylor Woodruff
1934 School Science and Mathematics  
Such a curve with several illustrations is found in Hamilton and Kettle Graphs and Imaginaries*® and in a lecture Graphical Computation for High School Students and Teachers by M. J.  ...  The method of this algebra leads up to the geometrical solution of the quadratic equation.’  ... 
doi:10.1111/j.1949-8594.1934.tb10841.x fatcat:jqav7bgawrdori2gbzijsxqwae

How to Stay Socially Distant: A Geometric Approach [article]

Omrit Filtser, Mayank Goswami, Joseph S.B. Mitchell, Valentin Polishchuk
2022 arXiv   pre-print
We introduce the notion of social distance width (SDW) in geometric domains, to model and quantify the ability for two or more agents to maintain social distancing while moving within their respective,  ...  We draw connections between our proposed social distancing measure and existing related work in computational geometry, hoping that our new measure may spawn investigations into further interesting problems  ...  We draw connections between our proposed social distancing measure and existing related work in computational geometry, hoping that our new measure may spawn investigations into further interesting problems  ... 
arXiv:2203.04548v1 fatcat:iv42zsg5kbcwxmbrmzininpobu

A story of diameter, radius and Helly property [article]

Feodor F. Dragan, Guillaume Ducoffe
2019 arXiv   pre-print
These above results are a new step toward better understanding the role of abstract geometric properties in the fast computation of metric graph invariants.  ...  graphs with constant VC-dimension the diameter can be computed in truly subquadratic time.  ...  graph runs in time quadratic in the number of elements in the ground set of the input family.  ... 
arXiv:1910.10412v2 fatcat:qe4v3hal75dvvpysjn5rcpq3hm

Geometric constraints in protein folding [article]

Nora Molkenthin, Steffen Mühle, Antonia S J S Mey, Marc Timme
2018 arXiv   pre-print
We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological  ...  These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.  ...  As described in the method section, the process of moving spheres towards each other is realized in a simple consistent way to satisfy all geometric constraints continuously in time.  ... 
arXiv:1812.09692v1 fatcat:dz2oilq4cfbwdla5n6up6i5bra

Confocal Stereo [chapter]

Samuel W. Hasinoff, Kiriakos N. Kutulakos
2006 Lecture Notes in Computer Science  
We present confocal stereo, a new method for computing 3D shape by controlling the focus and aperture of a lens.  ...  The method is specifically designed for reconstructing scenes with high geometric complexity or fine-scale texture.  ...  ., the scene surface intersected by C xy (α, f ), which varies in general with lens setting.  ... 
doi:10.1007/11744023_48 fatcat:mnidj6cjfvdi3f4lcpxtcscb5a

Confocal Stereo

Samuel W. Hasinoff, Kiriakos N. Kutulakos
2008 International Journal of Computer Vision  
We present confocal stereo, a new method for computing 3D shape by controlling the focus and aperture of a lens.  ...  The method is specifically designed for reconstructing scenes with high geometric complexity or fine-scale texture.  ...  ., the scene surface intersected by C xy (α, f ), which varies in general with lens setting.  ... 
doi:10.1007/s11263-008-0164-2 fatcat:px4nxn5trjc6tnhf2n7i5ciikm

Computing Voronoi skeletons of a 3-D polyhedron by space subdivision

Michal Etzion, Ari Rappoport
2002 Computational geometry  
; (2) the approximate Voronoi graph (AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the proximity structure diagram (PSD), which enhances the VG  ...  Second, the algorithm enables purely local computation of the skeletons in any limited region of interest.  ...  Computation of the PSD is very stable, since it does not involve symbolic decisions, and it utilizes the same simple geometric operations used in the computation of the Voronoi graph.  ... 
doi:10.1016/s0925-7721(01)00056-6 fatcat:d3wrf4zrsnakbidrj3d44srboe

Geometric constraints in protein folding [article]

Nora Molkenthin, Steffen Muehle, Antonia Mey, Marc Timme
2018 bioRxiv   pre-print
We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological  ...  These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.  ...  overdamped quadratic in the distances , = ‖ − ‖.  ... 
doi:10.1101/504399 fatcat:kjetyooinfebtfdtn3dgrt4kdi

One Rule to Grow Them All: A General Theory of Neuronal Branching and Its Practical Application

Hermann Cuntz, Friedrich Forstner, Alexander Borst, Michael Häusser, Abigail Morrison
2010 PLoS Computational Biology  
Additions to this rule, when required in the construction process, can be directly attributed to developmental processes or a neuron's computational role within its neural circuit.  ...  Here we propose such a formalism, which is derived from the expression of dendritic arborizations as locally optimized graphs.  ...  Farrow, Yihwa Kim, Philipp Rautenberg, Martin O'Reilly and Sarah Rieubland for testing parts of the TREES toolbox software package; Kate Buchanan and Jesper Sjöström for providing the filled neurons in  ... 
doi:10.1371/journal.pcbi.1000877 pmid:20700495 pmcid:PMC2916857 fatcat:wv3k2v4uznhpjnlmhqillh2xj4

Ad-hoc networks beyond unit disk graphs

Fabian Kuhn, Aaron Zollinger
2003 Proceedings of the 2003 joint workshop on Foundations of mobile computing - DIALM-POMC '03  
We prove that in Quasi Unit Disk Graphs flooding is an asymptotically message-optimal routing technique, provide a geometric routing algorithm being more efficient above all in dense networks, and show  ...  that classic geometric routing is possible with the same performance guarantees as for Unit Disk Graphs if d ≥ 1/ √ 2.  ...  In Section 8, we extend this result towards efficiency. Theorem 6.1.  ... 
doi:10.1145/941079.941089 dblp:conf/dialm/KuhnZ03 fatcat:u4uax5dx55dxxnyx4ggvehto6i

A survey on Mesh Segmentation Techniques

Ariel Shamir
2008 Computer graphics forum (Print)  
Recently, these have become a part of many mesh and object manipulation algorithms in computer graphics, geometric modelling and computer aided design.  ...  For instance, maximum or minimum ratio of diameter or perimeter to the area of the sub-mesh can provide a bias towards compact, roundly shaped sub-meshes.  ...  Some typical geometric constraints are: r Maximum/minimum area of sub-mesh, r Maximum/minimum length of diameter or perimeter of sub-mesh, r More complex constraints such as convexity of either 2D patch  ... 
doi:10.1111/j.1467-8659.2007.01103.x fatcat:wvrtpb642jhyxe7ikosjrm5qtu

Ad hoc networks beyond unit disk graphs

Fabian Kuhn, Roger Wattenhofer, Aaron Zollinger
2007 Wireless networks  
We prove that in Quasi Unit Disk Graphs flooding is an asymptotically message-optimal routing technique, provide a geometric routing algorithm being more efficient above all in dense networks, and show  ...  that classic geometric routing is possible with the same performance guarantees as for Unit Disk Graphs if d ≥ 1/ √ 2.  ...  In Section 8, we extend this result towards efficiency. Theorem 6.1.  ... 
doi:10.1007/s11276-007-0045-6 fatcat:waov4bfairghtjhxpzxogyksru
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