The labeling of point features on a map is a well-studied topic. In a static setting, the goal is to find a non-overlapping label placement for (a subset of) point features. In a dynamic setting, the set of point features and their corresponding labels change, and the labeling has to adapt to such changes. To aid the user in tracking these changes, we can use morphs, here called transitions, to indicate how a labeling changes. Such transitions have not gained much attention yet, and we investigate different types of transitions for labelings of points, most notably consecutive transitions and simultaneous transitions. We give (tight) upper bounds on the number of overlaps that can occur during these transitions. When each label has a non-negative weight associated to it, and each overlap imposes a penalty proportional to the weight of the overlapping labels, we show that it is NP-complete to decide whether the penalty during a simultaneous transition has weight at most $k$. Finally, we consider geotagged data on a map, by labeling points with rectangular or square labels. We developed a prototype implementation to evaluate different transition styles in practice, measuring both number of overlaps and transition duration.

翻译：地图上点特征的标注是一个经过充分研究的课题。在静态设置中，目标是找到点特征（或其子集）的无重叠标注布局。在动态设置中，点特征集合及其对应标注会发生变化，标注必须适应此类变化。为帮助用户跟踪这些变化，我们可以使用形变（此处称为过渡）来指示标注如何变化。此类过渡尚未引起广泛关注，我们研究了点标注的不同过渡类型，特别是连续过渡与同步过渡。我们给出了这些过渡过程中可能发生的重叠数量的（紧）上界。当每个标注具有非负权重，且每次重叠产生与重叠标注权重成比例的惩罚时，我们证明了判定同步过渡过程中的惩罚权重是否不超过$k$是NP完全问题。最后，我们通过使用矩形或正方形标注对点进行标注，研究地图上的地理标记数据。我们开发了原型实现来评估实际应用中不同过渡风格的表现，同时测量重叠数量和过渡持续时间。