Consistent Labeling of Rotating Maps

TL;DR

This paper addresses the problem of consistently labeling rotating maps to maximize visible labels, introducing models, proving NP-completeness, and providing approximation algorithms including an EPTAS for practical solutions.

Contribution

It models the dynamic labeling problem for rotating maps, proves its NP-completeness, and develops approximation algorithms including an efficient polynomial-time approximation scheme.

Findings

NP-completeness of active interval maximization even for unit-square labels

A constant-factor approximation based on line stabbing

An efficient polynomial-time approximation scheme (EPTAS) for the problem

Abstract

Dynamic maps that allow continuous map rotations, e.g., on mobile devices, encounter new issues unseen in static map labeling before. We study the following dynamic map labeling problem: The input is a static, labeled map, i.e., a set P of points in the plane with attached non-overlapping horizontal rectangular labels. The goal is to find a consistent labeling of P under rotation that maximizes the number of visible labels for all rotation angles such that the labels remain horizontal while the map is rotated. A labeling is consistent if a single active interval of angles is selected for each label such that labels neither intersect each other nor occlude points in P at any rotation angle. We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NP-completeness of the active interval maximization 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). Finally, we extend the EPTAS to the more general setting of rectangular labels of bounded size and aspect ratio.