A Note on Parallel Algorithms for Optional h-v Drawings of Binary Trees

In this paper we present a method to obtain optimal h-v drawings in parallel. Based on parallel tree contraction, our method computes optimal (with respect to a class of cost functions of the enclosing rectangle) drawings in O(log2 n) parallel time by using a polynomial number of EREW processors. The number of processors reduces substantially when we study minimum area drawings. Our work places the problem of obtaining optimal size h-v drawings in NC, presenting the first algorithm with polylogarithmic time complexity

L-Drawings of Directed Graphs

We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of matrix representations. In an L-drawing, vertices have exclusive $x$- and $y$-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-completeness and provide a heuristics based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. We performed an experimental analysis of the heuristics which confirms its effectiveness.Comment: 11 pages, 7 figure

Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

Snapping Graph Drawings to the Grid Optimally

In geographic information systems and in the production of digital maps for small devices with restricted computational resources one often wants to round coordinates to a rougher grid. This removes unnecessary detail and reduces space consumption as well as computation time. This process is called snapping to the grid and has been investigated thoroughly from a computational-geometry perspective. In this paper we investigate the same problem for given drawings of planar graphs under the restriction that their combinatorial embedding must be kept and edges are drawn straight-line. We show that the problem is NP-hard for several objectives and provide an integer linear programming formulation. Given a plane graph G and a positive integer w, our ILP can also be used to draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Sunspot areas and tilt angles for solar cycles 7-10

Extending the knowledge about the properties of solar cycles into the past is essential for understanding the solar dynamo. This paper aims at estimating areas of sunspots observed by Schwabe in 1825-1867 and at calculating the tilt angles of sunspot groups. The sunspot sizes in Schwabe's drawings are not to scale and need to be converted into physical sunspot areas. We employed a statistical approach assuming that the area distribution of sunspots was the same in the 19th century as it was in the 20th century. Umbral areas for about 130,000 sunspots observed by Schwabe were obtained, as well as the tilt angles of sunspot groups assuming them to be bipolar. There is, of course, no polarity information in the observations. The annually averaged sunspot areas correlate reasonably with sunspot number. We derived an average tilt angle by attempting to exclude unipolar groups with a minimum separation of the two alleged polarities and an outlier rejection method which follows the evolution of each group and detects the moment it turns unipolar at its decay. As a result, the tilt angles, although displaying considerable scatter, place the leading polarity on average 5.85+-0.25 closer to the equator, in good agreement with tilt angles obtained from 20th-century data sets. Sources of uncertainties in the tilt angle determination are discussed and need to be addressed whenever different data sets are combined. The sunspot area and tilt angle data are provided online.Comment: accepted for publication in Astron. & Astrophy

Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018