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Exact Algorithms for the Max-Min Dispersion Problem

  • Toshihiro Akagi
  • Tetsuya Araki
  • Takashi Horiyama
  • Shin-ichi NakanoEmail author
  • Yoshio Okamoto
  • Yota Otachi
  • Toshiki Saitoh
  • Ryuhei Uehara
  • Takeaki Uno
  • Kunihiro Wasa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10823)

Abstract

Given a set P of n elements, and a function d that assigns a non-negative real number d(pq) for each pair of elements \(p,q \in P\), we want to find a subset \(S\subseteq P\) with \(|S|=k\) such that \(\mathop {\mathrm {cost}}(S)= \min \{ d(p,q) \mid p,q \in S\}\) is maximized. This is the max-min k-dispersion problem. In this paper, exact algorithms for the max-min k-dispersion problem are studied. We first show the max-min k-dispersion problem can be solved in \(O(n^{\omega k/3} \log n)\) time. Then, we show two special cases in which we can solve the problem quickly. Namely, we study the cases where a set of n points lie on a line and where a set of n points lie on a circle (and the distance is measured by the shortest arc length on the circle). We obtain O(n)-time algorithms after sorting.

Keywords

Dispersion problem Algorithm 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Toshihiro Akagi
    • 1
  • Tetsuya Araki
    • 2
  • Takashi Horiyama
    • 3
  • Shin-ichi Nakano
    • 1
    Email author
  • Yoshio Okamoto
    • 4
    • 5
  • Yota Otachi
    • 6
  • Toshiki Saitoh
    • 7
  • Ryuhei Uehara
    • 8
  • Takeaki Uno
    • 9
  • Kunihiro Wasa
    • 9
  1. 1.Gunma UniversityKiryuJapan
  2. 2.Tokyo Metropolitan UniversityHachiojiJapan
  3. 3.Saitama UniversitySaitamaJapan
  4. 4.The University of Electro-CommunicationsChofuJapan
  5. 5.RIKEN Center for Advanced Intelligence ProjectTokyoJapan
  6. 6.Kumamoto UnivesityKumamotoJapan
  7. 7.Kyushu Institute of TechnologyKitakyushuJapan
  8. 8.JAISTNomiJapan
  9. 9.National Institute of InformaticsTokyoJapan

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