Optimal Algorithm for a Special Point-Labeling Problem

Abstract

We investigate a special class of map labeling problem. Let P = {p ₁, p ₂,..., p _n} be a set of point sites distributed on a 2D map. A label associated with each point is a axis-parallel rectangle of a constant height but of variable width. Here height of a label indicates the font size and width indicates the number of characters in that label. For a point p _i, its label contains the point p _i at its top-left or bottom-left corner, and it does not obscure any other point in P. Width of the label for each point in P is known in advance. The objective is to label the maximum number of points on the map so that the placed labels are mutually nonoverlapping. We first consider a simple model for this problem. Here, for each point p _i, the corner specification (i.e., whether the point p _i would appear at the top-left or bottom-left corner of the label) is known. We formulate this problem as finding the maximum independent set of a chordal graph, and propose an O(nlogn) time algorithm for producing the optimal solution. If the corner specification of the points in P is not known, our algorithm is a 2-approximation algorithm. Next, we develop a good heuristic algorithm that is observed to produce optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in [13].