Labeling Points with Weights

Abstract

Annotating maps, graphs, and diagrams with pieces of text is an important step in information visualization that is usually referred to as label placement. We define nine label-placement models for labeling points with axis-parallel rectangles given a weight for each point. There are two groups: fixed-position models and slider models. We aim to maximize the weight sum of those points that receive a label.We first compare our models by giving bounds for the ratios between the weights of maximum-weight labelings in different models. Then we present algorithms for labeling n points with unit-height rectangles. We show how an O(n log n)-time factor-2 approximation algorithm and a PTAS for fixed-position models can be extended to handle the weighted case. Our main contribution is the first algorithm for weighted sliding labels. Its approximation factor is (2 + ε), it runs in O(n 2 /ε) time and uses O(n/ε) space. We show that other than for fixed-position models even the projection to one dimension remains NP-hard.For slider models we also investigate some special cases, namely (a) the number of different point weights is bounded, (b) all labels are unit squares, and (c) the ratio between maximum and minimum label height is bounded.

Two Polynomial Time Algorithms for the Metro-line Crossing Minimization Problem

“…The map labeling problem has also attracted the interest of several researchers. Even if the majority of map labeling problems are shown to be N P -complete [7,9,10,13], several approaches have been suggested, among them expert systems [2], approximation algorithms [1,7,13,15], zero-one integer programming [17], simulated annealing [18]. An extensive bibliography on map labeling is maintained by Strijk and Wolff [16].…”

Maximizing the Number of Independent Labels in the Plane

“…In 1998, Agarwal et al [1] provided a (1 + 1/k)-factor algorithm that runs in O(n log n + n 2k−1 ) time, for any integer k ≥ 1, for fixed-height rectangle label placement model in the plane and an O(log n)-factor approximation algorithm that runs in O(n log n) time for arbitrary rectangle labels. Poon et al [19] further considered the weighted case in which each label is associated with a given weight and provided the same approximation result for 4P fixed-height weighted rectangle model. They also gave a (2 + )-factor approximation algorithm that runs in O(n 2 / ) time for 1d4S weighted rectangle label.…”

“…As a contrast to previous related results [1,3] in which the maximum independent set of label placement problem in the plane was considered and polynomial time approximation schemes (PTAS) were provided using the line stabbing technique and the shifting idea, we present a faster approach based on a different form of dynamic programming strategy and a particular analysis to solving this Max-1d4P model in O(n log Δ) time which improves previously known results that run in O(n 2 ) and O(nΔ) time in the worse case. We also point out an implicit difference between point labeling problem and label placement problem, mentioned in the intuitive proof of the reduction [19]. In addition, we further extend our method to solve the fixed-height rectangle label placement model in the plane and present a (1 + 1/k)-factor polynomial time approximation scheme (PTAS) algorithm that runs in O(n log n + kn log 4 Δ + Δ k−1 ) time, using O(kΔ 3 …”

“…Since this problem in general is known to be NP-complete, many heuristics or special cases for which polynomial time algorithms are given have been presented. For instance, there are many algorithms that have been developed for labeling points that are on lines [5,9,11,12,19,22] or in a region [7,8,13,14,15,16,17,18,20,21].…”

“…This problem has been studied previously [9,16,19,5]. The prefix 1d or Slope refers to the problem in which the anchors lie on a horizontal or a sloping line, respectively.…”