Abstract
We consider map labeling for the case that a map undergoes a sequence of operations such as rotation, zoom and translation over a specified time span. We unify and generalize several previous models for dynamic map labeling into one versatile and flexible model. In contrast to previous research, we completely abstract from the particular operations and express the labeling problem as a set of time intervals representing the labels’ presences, activities and conflicts. One of the model’s strength is manifested in its simplicity and broad range of applications. In particular, it supports label selection both for map features with fixed position as well as for moving entities (e.g., for tracking vehicles in logistics or air traffic control). We study the active range maximization problem in this model. We prove that the problem is NP-complete and W[1]-hard, and present constant-factor approximation algorithms. In the restricted, yet practically relevant case that no more than k labels can be active at any time, we give polynomial-time algorithms as well as constant-factor approximation algorithms.
Highlights
Dynamic digital maps are becoming more and more ubiquitous, especially with the rising numbers of location-based services and smartphone users worldwide
We introduced a temporal model for dynamic map labeling that satisfies the consistency criteria demanded by Been et al [1], even in a stronger sense, where each activity change of a label must be explainable to the user by some witness label
Our model transforms the geometric information specified by the motion of the camera as well as the labels into the two combinatorial problems GeneralMaxTotal and k-RestrictedMaxTotal that are expressed in terms of presence and conflict intervals
Summary
Dynamic digital maps are becoming more and more ubiquitous, especially with the rising numbers of location-based services and smartphone users worldwide. Gemsa et al [10, 11] extend the ARO model to rotation operations They first show that the ARO problem is NP-hard in the considered setting and introduce an efficient polynomial-time-approximation scheme (FPTAS) for unit-height rectangles [10]. Depending on the objective and additional consistency constraints of the labeling model, different sets of subintervals may be chosen by the algorithm This is a very versatile model, which includes, for instance, map labeling for car navigation systems, in which the map view changes position, angle, and scale according to the car’s position, heading, and speed following a particular route; see Fig. 2. On we assume that an activity set is valid, unless stated otherwise
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