Labeling Points with Rectangles of Various Shapes

We deal with a map-labeling problem, named LOFL (Leftpart Ordered Flexible Labeling), to label a set of points in a plane with polygonal obstacles. The label for each point is selected from a set of rectangles with various shapes satisfying the left-part ordered property, and is placed near to the point after scaled by a scaling factor σ which is common to all points. In this paper, we give an optimal O((n+m) log(n+ m)) algorithm to decide the feasibility of LOFL for a fixed scaling factor σ, and an O((n + m) log(n + m)) time algorithm to find the largest feasible scaling factor σ, where n is the number of points and m is the number of edges of polygonal obstacles.