Time-Space Trade-offs for Triangulations and Voronoi Diagrams

  • Matias Korman
  • Wolfgang Mulzer
  • André van Renssen
  • Marcel Roeloffzen
  • Paul Seiferth
  • Yannik Stein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in \(O(n \log n)\) time and O(n) space. We study the situation when the available workspace is limited: given a parameter \(s \in \{1, \dots , n\}\), an s-workspace algorithm has read-only access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of \(\Theta (\log n)\) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing a triangulation of S in time \(O(n^2/s + n \log n \log s )\) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time \(O((n^2/s) \log s + n \log s \log ^*s)\).

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Matias Korman
    • 1
  • Wolfgang Mulzer
    • 2
  • André van Renssen
    • 1
  • Marcel Roeloffzen
    • 3
  • Paul Seiferth
    • 2
  • Yannik Stein
    • 2
  1. 1.JST, ERATO, Kawarabayashi Large Graph ProjectNational Institute of Informatics (NII)TokyoJapan
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Tohoku UniversitySendaiJapan

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