4 Search Results for "Basit, Abdul"

Document
DOI: 10.4230/LIPIcs.ESA.2025.67
Compact Representation of Semilinear and Terrain-Like Graphs

Authors: Jean Cardinal and Yelena Yuditsky

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994). Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.
Cite as

Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2025.67,
  author =	{Cardinal, Jean and Yuditsky, Yelena},
  title =	{{Compact Representation of Semilinear and Terrain-Like Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.67},
  URN =		{urn:nbn:de:0030-drops-245359},
  doi =		{10.4230/LIPIcs.ESA.2025.67},
  annote =	{Keywords: Biclique covers, intersection graphs, visibility graphs, Zarankiewicz’s problem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.17
Sparse Bounded Hop-Spanners for Geometric Intersection Graphs

Authors: Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n). Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}).
Cite as

Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth. Sparse Bounded Hop-Spanners for Geometric Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2025.17,
  author =	{Bhore, Sujoy and Chan, Timothy M. and Huang, Zhengcheng and Smorodinsky, Shakhar and T\'{o}th, Csaba D.},
  title =	{{Sparse Bounded Hop-Spanners for Geometric Intersection Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.17},
  URN =		{urn:nbn:de:0030-drops-231698},
  doi =		{10.4230/LIPIcs.SoCG.2025.17},
  annote =	{Keywords: Geometric Spanners, Geometric Intersection Graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.8
Eight-Partitioning Points in 3D, and Efficiently Too

Authors: Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner

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

Abstract
An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).
Cite as

Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner. Eight-Partitioning Points in 3D, and Efficiently Too. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aronov_et_al:LIPIcs.SoCG.2024.8,
  author =	{Aronov, Boris and Basit, Abdul and Ramesh, Indu and Tasinato, Gianluca and Wagner, Uli},
  title =	{{Eight-Partitioning Points in 3D, and Efficiently Too}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{8:1--8:15},
  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.2024.8},
  URN =		{urn:nbn:de:0030-drops-199538},
  doi =		{10.4230/LIPIcs.SoCG.2024.8},
  annote =	{Keywords: Mass partitions, partitions of points in three dimensions, Borsuk-Ulam Theorem, Ham-Sandwich Theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.15
On the Number of Ordinary Lines Determined by Sets in Complex Space

Authors: Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.
Cite as

Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf. On the Number of Ordinary Lines Determined by Sets in Complex Space. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{basit_et_al:LIPIcs.SoCG.2017.15,
  author =	{Basit, Abdul and Dvir, Zeev and Saraf, Shubhangi and Wolf, Charles},
  title =	{{On the Number of Ordinary Lines Determined by Sets in Complex Space}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.15},
  URN =		{urn:nbn:de:0030-drops-71883},
  doi =		{10.4230/LIPIcs.SoCG.2017.15},
  annote =	{Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method}
}
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