Map labeling problems

Abstract

One of the most challenging tasks in cartography is the labeling of maps - attaching text to geographic features. Many of the issues become simpler when the features are points, for example cities on a large-scale map, because we expect the text to be placed horizontally and close to the associated point. We want labels that do not overlap and are large enough to be readable. Even simple formulations of this problem are NP-complete. One such formulation is the point-feature label placement problem: given a set of points in the plane, and an axis-parallel rectangular label associated with each point, place each label with one corner at the associated point such that no two labels overlap. This problem is known to be NP-complete.

Modeling each label as a fixed rectangle in this way is quite limiting. Researchers have considered approximation algorithms where each label can be scaled.

In this thesis, we propose an alternative formulation to the map labeling problem. We introduce the use of elastic labels, where each label is a rectangle with fixed area, but varying in height and width. Then, we define the elastic labeling problem as determining the choice of height and width of each label, and the corner of the label to place at the associated point, so that no two labels overlap. This problem is useful when the goal of placing a label at a given point is to associate some text, consisting of more than one word, with the point. In this case we can write the specified text inside the label using one, two, or more rows, as long as the label is placed at a specified point.

The elastic labeling problem is an NP-hard problem, since it generalizes the point-feature label placement problem. That is, the problem is NP-hard even when there is no elasticity just because of the choice of the corners. We show that the problem remains NP-hard when we have elasticity but no choice about which corner of the label to use. We call this the one-corner elastic labeling problem.

Thus, we consider a problem with more constraints, yet of practical use. We require that the points lie on the boundary of a rectangular map. This rectangle perimeter labeling problem arises, for example, when the perimeter of the map is labeled with information about objects that lie beyond the boundary of the map, e.g. where the roads lead to. This problem is likely to be relevant in Geographical Information Systems (GIS) as maps are displayed dynamically on a computer screen using clipping, panning, and zooming. It is also applicable for labeling the coordinate axes in graphs such as charts or histograms.

Our main result is a polynomial time algorithm for the rectangle perimeter labeling problem. To solve the rectangle perimeter labeling problem, we look at two problems. The first problem is when the points lie on two adjacent sides of the rectangular map, the two-axis labeling problem. The second problem is when the points lie on two opposite sides of the rectangular map, the two-parallel lines labeling problem.

In this thesis, we also study another variant of the point-feature label placement problem: the sliding labels problem. In this problem each label has fixed dimension but any boundary point (not just a corner) of the rectangular label can be placed at the associated point in the plane. This problem is known to be NP-hard. We show that it remains NP-hard when points can only be placed on the left or right sides of the boundary of their associated labels - we call this the left-right sliding labels problem.

As well we give algorithms for special cases of the elastic labeling problem and the sliding labels problem. Finally, we study approximation algorithms for the point-feature label placement problem.