[서평]「The Unified Modeling Language User Guide」

Computational geometry

Quantized consensus

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

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.

/pdf/boundary-labeling-models-and-efficient-algorithms-for-1nm8nf4fra.pdf

Boundary labeling: Models and efficient algorithms for rectangular maps

This paper presents the problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that $\Omega (\frac{1}{{d^2 }}) \leqslant R \leqslant \frac{{2\pi }}{d}$ for any graph. Moreover, it is proved that $R = \Theta (\frac{1}{d})$ for many graphs including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networks. It is also shown that the problem of deciding if $R = \frac{{2\pi }}{d}$ for a graph is NP-hard for $d = 4$, and by using a counting argument that $R = O(\frac{{\log d}}{{d^2 }})$ for many graphs.

Drawing graphs in the plane with high resolution

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.

The Straight-Line RAC Drawing Problem is NP-Hard

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.

/pdf/boundary-labeling-models-and-efficient-algorithms-for-1zjhpczfpq.pdf

Boundary labeling: models and efficient algorithms for rectangular maps

https://cs.mcgill.ca/~sue/papers/eades-sym-whi.pdf

Antonios Symvonis

Papers

Boundary labeling: Models and efficient algorithms for rectangular maps

Drawing graphs in the plane with high resolution

The Straight-Line RAC Drawing Problem is NP-Hard

Boundary labeling: models and efficient algorithms for rectangular maps

Three-dimensional orthogonal graph drawing algorithms