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Balanced Line Separators of Unit Disk Graphs

  • Paz Carmi
  • Man Kwun Chiu
  • Matthew J. Katz
  • Matias Korman
  • Yoshio Okamoto
  • André van Renssen
  • Marcel Roeloffzen
  • Taichi Shiitada
  • Shakhar Smorodinsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line \(\ell \) such that \(\ell \) intersects at most \(O(\sqrt{(m+n)\log {n}})\) disks and each of the halfplanes determined by \(\ell \) contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting \(O(\sqrt{m+n})\) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when \(m=0\). Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size \(O(\sqrt{m})\)).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Paz Carmi
    • 1
  • Man Kwun Chiu
    • 2
    • 3
  • Matthew J. Katz
    • 1
  • Matias Korman
    • 4
  • Yoshio Okamoto
    • 5
  • André van Renssen
    • 2
    • 3
  • Marcel Roeloffzen
    • 2
    • 3
  • Taichi Shiitada
    • 5
  • Shakhar Smorodinsky
    • 1
  1. 1.Ben-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.National Institute of InformaticsTokyoJapan
  3. 3.JST, ERATO, Kawarabayashi Large Graph ProjectTokyoJapan
  4. 4.Tohoku UniversitySendaiJapan
  5. 5.The University of Electro-CommunicationsTokyoJapan

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