Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.-We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. -We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). -We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.
In order to accomplish cost-efficient management of complex optical communication networks, operators are seeking automation of network diagnosis and management by means of Machine Learning (ML). To support these objectives, new functions are needed to enable cognitive, autonomous management of optical network security. This paper focuses on the challenges related to the performance of ML-based approaches for detection and localization of optical-layer attacks, and to their integration with standard Network Management Systems (NMSs). We propose a framework for cognitive security diagnostics that comprises an attack detection module with Supervised Learning (SL), Semi-Supervised Learning (SSL) and Unsupervised Learning (UL) approaches, and an attack localization module that deduces the location of a harmful connection and/or a breached link. The influence of false positives and false negatives is addressed by a newly proposed Window-based Attack Detection (WAD) approach. We provide practical implementation guidelines for the integration of the framework into the NMS and evaluate its performance in an experimental network testbed subjected to attacks, resulting with the largest optical-layer security experimental dataset reported to date.
It is well known that any graph admits a crossing-free straight-line drawing in R 3 and that any planar graph admits the same even in R 2 . For a graph G and d ∈ {2, 3}, let ρ 1 d (G) denote the minimum number of lines in R d that together can cover all edges of a drawing of G. For d = 2, G must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. *
Abstract. Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an xmonotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally.
Abstract:The force-directed paradigm is one of the few generic approaches to drawing graphs. Since force-directed algorithms can be extended easily, they are used frequently. Most of these algorithms are, however, quite slow on large graphs, as they compute a quadratic number of forces in each iteration. We give a new algorithm that takes only O(m + n log n) time per iteration when laying out a graph with n vertices and m edges. Our algorithm approximates the true forces using the so-called well-separated pair decomposition. We perform experiments on a large number of graphs and show that we can strongly reduce the runtime, even on graphs with less than a hundred vertices, without a significant influence on the quality of the drawings (in terms of the number of crossings and deviation in edge lengths).
No abstract
Storyline visualizations show the structure of a story, by depicting the interactions of the characters over time. Each character is represented by an x-monotone curve from left to right, and a meeting is represented by having the curves of the participating characters run close together for some time. There have been various approaches to drawing storyline visualizations in an automated way. In order to keep the visual complexity low, rather than minimizing pairwise crossings of curves, we count block crossings, that is, pairs of intersecting bundles of lines. Partly inspired by the ILP-based approach of Gronemann et al. [GD 2016] for minimizing the number of pairwise crossings, we model the problem as a satisfiability problem (since the straightforward ILP formulation becomes more complicated and harder to solve). Having restricted ourselves to a decision problem, we can apply powerful SAT solvers to find optimal drawings in reasonable time. We compare this SAT-based approach with two exact algorithms for block crossing minimization, using both the benchmark instances of Gronemann et al. and random instances. We show that the SAT approach is suitable for real-world instances and identify cases where the other algorithms are preferable.
We present a tool for annotating Latex documents with comments. Our annotations are placed in the left, right, or both margins, and connected to the corresponding positions in the text with arrows (so-called leaders). Problems of this type have been studied under the name boundary labeling. We consider various leader types (straight-line, rectilinear, and Bézier) and modify existing algorithms to allow for annotations of varying height. We have implemented our algorithms in Lua; they are available for download as an easy-to-use Luatex package. IntroductionMany word processing systems support annotations for the text. The most common case for this annotations are comments, which can be inserted in arbitrary positions inside the text. The comments themselves are placed as labels in the margin next to the text and connected to the corresponding position, called site, by a line called leader. The endpoint of a leader at a label is called a port. Such comments are available, for example, in LibreOffice (see Fig. 1) and Microsoft Word. This task can be expressed in the boundary labeling notion introduced by Bekos et al. [5]: the sites to be annotated lie inside the text area and the labels are to be placed outside the text area. They describe several types of leaders, such as straight-line leaders (s-leaders), rectilinear leaders with one bend (po-leaders) and rectilinear leaders with two bends (opo-leaders).Previous work. Boundary labeling has been extensively investigated in the last few years, see a survey on the interaction between cartography and graph drawing [17]. For labels of uniform size, the problem is well-studied. Most algorithms try to minimize the total leader length. For s-leaders, it suffices to compute a minimum-weight perfect matching, which can be done in O(n 2+ε ) time [1]. For opo-leaders, Bekos et al. [5] gave three different algorithms for the number of sides used by the labels, with running times O(n log n) (one-sided), O(n 2 ) (two-sided), and O(n 2 log 3 n) (four-sided). Further, they presented an O(n 2 )-time algorithm for po-leaders that lie on one side or on two opposite sides of the text. The result for po-leaders was improved by Benkert et al. [6] for the one-sided case. They gave an O(n log n)-time algorithm for length minimization and an O(n 3 )-time algorithm for a very general class of objective functions, including, for example,
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