Label placement plays an important role in information visualization. On maps, diagrams, and graphs, properly placed text labels clarify the meaning of a point, a line, or an area. It is known that placing labels is a costly part in a map making process. Many researchers in cartography and computational geometry have suggested various models and solutions.
In spite of the importance, most label placement problems are proved NP-hard. Therefore, one should consider an approximate solution instead of exact solution. An approximation algorithm guarantees that its solution is always near the exact solution. For example, in maximization problem, ε-approximation algorithm guarantees that the solution is greater than the exact solution over ε.(ε > 1)
The models of label placement defined can be classified by several ways: the feature to be labeled, the maximization objective, the shape of labels, the placement restriction, the number of labels for each feature, and so on.
In this thesis, we consider the size maximization problem of axis-parallel rectangular and square labels placed on points. First, we consider the following problem: Given n pairs of a point and an axis-parallel rectangle in the plane, place each rectangle at each point in order that the point lies on the corner of the rectangle and the rectangles do not intersect. If the size of the rectangles may be enlarged or reduced at the same factor, maximize the factor. Formann and Wagner showed that this problem is NP-complete and there is no polynomial time algorithm less than 2-approximation unless P = NP. We present a 2-pproximation algorithm of the non-uniform rectangle case which runs in $O(n^2log n)$ time and takes $O(n^2)$ space. We also show that the decision problem can be solved in O(nlog n) time and space.
We also propose a label placement model, called 8-position model, that the label can be placed among up, down, right, left, upper-right, upper-left, lower-right and lower left of the point....