3 Search Results for "Abam, Mohammad Ali"

Document
DOI: 10.4230/LIPIcs.SWAT.2024.2
Local Spanners Revisited

Authors: Stav Ashur and Sariel Har-Peled

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
For a set P ⊆ ℝ² of points and a family ℱ of regions, a local t-spanner of P is a sparse graph G over P, such that for any region r ∈ ℱ the subgraph restricted to r, denoted by G ∩ r, is a t-spanner for all the points of r ∩ P. We present algorithms for the construction of local spanners with respect to several families of regions such as homothets of a convex region. Unfortunately, the number of edges in the resulting graph depends logarithmically on the spread of the input point set. We prove that this dependency cannot be removed, thus settling an open problem raised by Abam and Borouny. We also show improved constructions (with no dependency on the spread) of local spanners for fat triangles, and regular k-gons. In particular, this improves over the known construction for axis-parallel squares. We also study notions of weaker local spanners where one is allowed to shrink the region a "bit". Surprisingly, we show a near linear-size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is multiplicative. Any spanner is a weak local spanner if the shrinking is proportional to the diameter of the region.
Cite as

Stav Ashur and Sariel Har-Peled. Local Spanners Revisited. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ashur_et_al:LIPIcs.SWAT.2024.2,
  author =	{Ashur, Stav and Har-Peled, Sariel},
  title =	{{Local Spanners Revisited}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.2},
  URN =		{urn:nbn:de:0030-drops-200420},
  doi =		{10.4230/LIPIcs.SWAT.2024.2},
  annote =	{Keywords: Geometric graphs, Fault-tolerant spanners}
}
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)

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@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}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.186
Geometric Spanners for Points Inside a Polygonal Domain

Authors: Mohammad Ali Abam, Marjan Adeli, Hamid Homapour, and Pooya Zafar Asadollahpoor

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function d where for any two points p and q, d(p,q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0), we construct a (sqrt(10)+eps)-spanner that has O(n log^2 n) edges where eps is the a given positive real number. For a case where there are h holes, our construction gives a (5+eps)-spanner with the size of O(sqrt(h) n log^2 n). Moreover, we study t-spanners for the visibility graph of P (VG(P), for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+eps)-spanner of size almost O(n^{4/3}). In addition, we show that there is a set P of n points such that any (3-eps)-spanner of VG(P) must contain almost n^2 edges.
Cite as

Mohammad Ali Abam, Marjan Adeli, Hamid Homapour, and Pooya Zafar Asadollahpoor. Geometric Spanners for Points Inside a Polygonal Domain. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 186-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{abam_et_al:LIPIcs.SOCG.2015.186,
  author =	{Abam, Mohammad Ali and Adeli, Marjan and Homapour, Hamid and Asadollahpoor, Pooya Zafar},
  title =	{{Geometric Spanners for Points Inside a Polygonal Domain}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{186--197},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.186},
  URN =		{urn:nbn:de:0030-drops-51378},
  doi =		{10.4230/LIPIcs.SOCG.2015.186},
  annote =	{Keywords: Geometric Spanners, Polygonal Domain, Visibility Graph}
}
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