Maximizing the number of independent labels in the plane

  • Authors:

  • Kuen-Lin Yu;Chung-Shou Liao;D. T. Lee

  • Affiliations:

  • Department of Computer Science and Information Engineering, National Chiao Tung University, HsinChu, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan and Institute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan and Institute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan

  • Venue:
  • FAW'07 Proceedings of the 1st annual international conference on Frontiers in algorithmics
  • Year:
  • 2007

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Abstract

In this paper, we consider a map labeling problem to maximize the number of independent labels in the plane. We first investigate the point labeling model that each label can be placed on a given set of anchors on a horizontal line. It is known that most of the map labeling decision models on a single line (horizontal or slope line) can be easily solved. However, the label number maximization models are more difficult (like 2SAT vs. MAX-2SAT). We present an O(n log Δ) time algorithm for the four position label model on a horizontal line based on dynamic programming and a particular analysis, where n is the number of the anchors and Δ is the maximum number of labels whose intersection is nonempty. As a contrast to Agarwal et al.'s result [Comput. Geom. Theory Appl. 11 (1998) 209-218] and Chan's result [Inform. Process. Letters 89(2004) 19-23] in which they provide (1 + 1/k)-factor PTAS algorithms that run in O(n log n + n2k-1) time and O(n log n + nΔk-1) time respectively for the fixed-height rectangle label placement model in the plane, we extend our method to improve their algorithms and present a (1 + 1/k)-factor PTAS algorithm that runs in O(n log n + kn log4 Δ + Δk-1) time using O(kΔ3 log4 Δ + kΔk-1) storage.