Michael A. Bekos

Boundary labeling: Models and efficient algorithms for rectangular maps

We introduce boundary labeling, a new model for labeling point sites with large labels. According to the boundary-labeling model, labels are placed around an axis-parallel rectangle that contains the point sites, each label is connected to its corresponding site through a polygonal line called leader, and no two leaders intersect. Although boundary labeling is commonly used, e.g., for technical drawings and illustrations in medical atlases, this problem has not yet been studied in the literature. The problem is interesting in that it is a mixture of a label-placement and a graph-drawing problem. In this paper we investigate several variants of the boundary-labeling problem. We consider labels of identical or different size, straight-line or rectilinear leaders, fixed or sliding ports for attaching leaders to sites and attaching labels to one, two or all four sides of the bounding rectangle. For any variant of the boundary labeling model, we aim at highly esthetical placements of labels and leaders. We present simple and efficient algorithms that minimize the total leader length or, in the case of rectilinear leaders, the total number of bends.

Boundary Labeling with Octilinear Leaders

A major factor affecting the readability of an illustration that contains textual labels is the degree to which the labels obscure graphical features of the illustration as a result of spatial overlaps. Boundary labeling addresses this problem by attaching the labels to the boundary of a rectangle that contains all features. Then, each feature should be connected to its associated label through a polygonal line, called leader, such that no two leaders intersect. In this paper we study the boundary labeling problem along a new line of research, according to which different pairs of type leaders (i.e. doand pd, odand pd) are combined to produce boundary labelings. Thus, we are able to overcome the problem that there might be no feasible solution when labels are placed on different sides and only one type of leaders is allowed. Our main contribution is a new algorithm for solving the total leader length minimization problem (i.e., the problem of finding a crossing free boundary labeling, such that the total leader length is minimized) assuming labels of uniform size. We also present an NP-completeness result for the case where the labels are of arbitrary size.

Boundary labeling: models and efficient algorithms for rectangular maps

In this paper, we present boundary labeling, a new approach for labeling point sets with large labels. We first place disjoint labels around an axis-parallel rectangle that contains the points. Then we connect each label to its point such that no two connections intersect. Such an approach is common e.g. in technical drawings and medical atlases, but so far the problem has not been studied in the literature. The new problem is interesting in that it is a mixture of a label-placement and a graph-drawing problem.

The straight-line RAC drawing problem is NP-hard

Recent cognitive experiments have shown that the negative impact of an edge crossing on the human understanding of a graph drawing, tends to be eliminated in the case where the crossing angles are greater than 70 degrees. This motivated the study of RAC drawings, in which every pair of crossing edges intersects at right angle. In this work, we demonstrate a class of graphs with unique RAC combinatorial embedding and we employ members of this class in order to show that it is NP-hard to decide whether a graph admits a straight-line RAC drawing.

Line crossing minimization on metro maps

We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway line which connects them, whereas the paths illustrate the lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM). In contrast to the problem of drawing the underlying graph nicely, MLCM has received fewer attention. It was recently introduced by Benkert et. al in [4]. In this paper, as a first step towards solving MLCM in arbitrary graphs, we study path and tree networks.We examine several variations of the problem for which we develop algorithms for obtaining optimal solutions.

On multi-stack boundary labeling problems

The boundary labeling problem was recently introduced in [5] as a response to the problem of labeling dense point sets with large labels. In boundary labeling, we are given a rectangle R which encloses a set of n sites. Each site is associated with an axis-parallel rectangular label. The main task is to place the labels in distinct positions on the boundary of R, so that they do not overlap, and to connect each site with its corresponding label by non-intersecting polygonal lines, so called leaders. Such a label placement is referred to as legal label placement. In this paper, we study boundary labeling problems along a new line of research. We seek to obtain labelings with labels arranged on more than one stacks placed at the same side of R. We refer to problems of this type as multi-stack boundary labeling problems. We present algorithms for maximizing the uniform label size for boundary labeling with two and three stacks of labels. The key component of our algorithms is a technique that combines the merging of lists and the bounding of the search space of the solution. We also present NP-hardness results for multi-stack boundary labeling problems with labels of variable height.

Area-Feature Boundary Labeling1

Boundary labeling is a relatively new labeling method. It can be useful in automating the production of technical drawings and medical drawings, where it is common to explain certain parts of the drawing with text labels, arranged on its boundary so that other parts of the drawing are not obscured. In boundary labeling, we are given a rectangle R which encloses a set of n sites. Each site s is associated with an axis-parallel rectangular label ls. The labels must be placed in distinct positions on the boundary of R and they must be connected to their corresponding sites with polygonal lines, called leaders, so that the labels are pairwise disjoint and the leaders do not intersect each other. In this paper, we study a version of the boundary labeling problem where the sites can ‘float’ within a polygonal region. We present a polynomial time algorithm, which runs in O(n3) time and produces a labeling of minimum total leader length for labels of uniform size placed in fixed positions on the boundary of rectangle R.

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends ?

Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is a drawing of the two graphs where each graph is drawn planar, no two edges overlap, and edges of one graph can cross edges of the other graph only at right angles. In the geometric version of the problem, vertices are drawn as points and edges as straight-line segments. It is known, however, that even pairs of very simple classes of planar graphs (such as wheels and matchings) do not always admit a geometric RAC simultaneous drawing.

On Metro-Line Crossing Minimization

We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V;E) so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway track connecting them, whereas the paths illustrate the metro lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM). We examine several variations of the problem for which we develop algorithms that yield optimal solutions.

On RAC drawings of 1-planar graphs

Abstract A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs.