Search Results

Documents authored by Saghafian, Morteza

Document
DOI: 10.4230/LIPIcs.GD.2024.3
The Euclidean MST-Ratio for Bi-Colored Lattices

Authors: Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, and Morteza Saghafian

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25.
Cite as

Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. The Euclidean MST-Ratio for Bi-Colored Lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{cultreradimontesano_et_al:LIPIcs.GD.2024.3,
  author =	{Cultrera di Montesano, Sebastiano and Draganov, Ond\v{r}ej and Edelsbrunner, Herbert and Saghafian, Morteza},
  title =	{{The Euclidean MST-Ratio for Bi-Colored Lattices}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.3},
  URN =		{urn:nbn:de:0030-drops-212878},
  doi =		{10.4230/LIPIcs.GD.2024.3},
  annote =	{Keywords: Minimum spanning Trees, Steiner Ratio, Lattices, Partitions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.76
Grid Peeling of Parabolas

Authors: Günter Rote, Moritz Rüber, and Morteza Saghafian

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.
Cite as

Günter Rote, Moritz Rüber, and Morteza Saghafian. Grid Peeling of Parabolas. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{rote_et_al:LIPIcs.SoCG.2024.76,
  author =	{Rote, G\"{u}nter and R\"{u}ber, Moritz and Saghafian, Morteza},
  title =	{{Grid Peeling of Parabolas}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{76:1--76:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.76},
  URN =		{urn:nbn:de:0030-drops-200213},
  doi =		{10.4230/LIPIcs.SoCG.2024.76},
  annote =	{Keywords: grid polygons, curvature flow}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.16
Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory

Authors: Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.
Cite as

Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{biswas_et_al:LIPIcs.SoCG.2021.16,
  author =	{Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  title =	{{Counting Cells of Order-k Voronoi Tessellations in \mathbb{R}³ with Morse Theory}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.16},
  URN =		{urn:nbn:de:0030-drops-138152},
  doi =		{10.4230/LIPIcs.SoCG.2021.16},
  annote =	{Keywords: Voronoi tessellations, Delaunay mosaics, arrangements, convex polytopes, Morse theory, counting}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2020.3
Preclustering Algorithms for Imprecise Points

Authors: Mohammad Ali Abam, Mark de Berg, Sina Farahzad, Mir Omid Haji Mirsadeghi, and Morteza Saghafian

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

Abstract
We study the problem of preclustering a set B of imprecise points in ℝ^d: we wish to cluster the regions specifying the potential locations of the points such that, no matter where the points are located within their regions, the resulting clustering approximates the optimal clustering for those locations. We consider k-center, k-median, and k-means clustering, and obtain the following results. Let B:={b₁,…,b_n} be a collection of disjoint balls in ℝ^d, where each ball b_i specifies the possible locations of an input point p_i. A partition 𝒞 of B into subsets is called an (f(k),α)-preclustering (with respect to the specific k-clustering variant under consideration) if (i) 𝒞 consists of f(k) preclusters, and (ii) for any realization P of the points p_i inside their respective balls, the cost of the clustering on P induced by 𝒞 is at most α times the cost of an optimal k-clustering on P. We call f(k) the size of the preclustering and we call α its approximation ratio. We prove that, even in ℝ^1, one may need at least 3k-3 preclusters to obtain a bounded approximation ratio - this holds for the k-center, the k-median, and the k-means problem - and we present a (3k,1) preclustering for the k-center problem in ℝ^1. We also present various preclusterings for balls in ℝ^d with d⩾2, including a (3k,α)-preclustering with α≈13.9 for the k-center and the k-median problem, and α≈254.7 for the k-means problem.
Cite as

Mohammad Ali Abam, Mark de Berg, Sina Farahzad, Mir Omid Haji Mirsadeghi, and Morteza Saghafian. Preclustering Algorithms for Imprecise Points. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{abam_et_al:LIPIcs.SWAT.2020.3,
  author =	{Abam, Mohammad Ali and de Berg, Mark and Farahzad, Sina and Mirsadeghi, Mir Omid Haji and Saghafian, Morteza},
  title =	{{Preclustering Algorithms for Imprecise Points}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{3:1--3:12},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.3},
  URN =		{urn:nbn:de:0030-drops-122503},
  doi =		{10.4230/LIPIcs.SWAT.2020.3},
  annote =	{Keywords: Geometric clustering, k-center, k-means, k-median, imprecise points, approximation algorithms}
}
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact