Maximizing the Number of Independent Labels in the Plane

Abstract

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. (1 + 1/k)-factor PTAS algorithms that run in O(n log n + 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 + 1/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.