Maximizing the Number of Independent Labels in the Plane

摘要

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 + l/k)-factor PTAS algorithms that run in O(nlogn + n~(2k-1)) time and O(n log n + nΔ~(k-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 + l/k)-factor PTAS algorithm that runs in O(n log n + kn log~4 Δ + Δ~(k-1)) time using O(kΔ ~3 log~4 Δ + kΔ~(k-1)) storage.