### Consistent labeling of rotating maps

Andreas Gemsa, Martin Nöllenburg, Ignaz Rutter

#### Abstract

Dynamic maps that allow continuous map rotations, for example, on mobile devices, encounter new geometric labeling issues unseen in static maps before. We study the following dynamic map labeling problem: The input is an abstract map consisting of a set $P$ of points in the plane with attached horizontally aligned rectangular labels. While the map with the point set $P$ is rotated, all labels remain horizontally aligned. We are interested in a consistent labeling of $P$ under rotation, i.e., an assignment of a single (possibly empty) active interval of angles for each label that determines its visibility under rotations such that visible labels neither intersect each other (soft conflicts) nor occlude points in $P$ at any rotation angle (hard conflicts). Our goal is to find a consistent labeling that maximizes the number of visible labels integrated over all rotation angles.

We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NP-hardness of the above optimization problem even for unit-square labels. We then present a constant-factor approximation for this problem based on line stabbing, and refine it further into an efficient polynomial-time approximation scheme (EPTAS).

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DOI: http://dx.doi.org/10.20382/jocg.v7i1a15

ISSN: 1920-180X