Smooth Orthogonal Layouts

Bekos, Michael A.; Kaufmann, Michael; Kobourov, Stephen G.; Symvonis, Antonios

doi:10.1007/978-3-642-36763-2_14

Cited by 13 publications

(43 citation statements)

References 25 publications

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“…orthogonal drawings and to improve their readability, Bekos et al [5] introduced smooth orthogonal drawings that combine the clarity of orthogonal layouts with the artistic style of Lombardi drawings [10] by replacing sequences of "hard" bends in the orthogonal drawing of the edges by (potentially shorter) sequences of "smooth" inflection points connecting circular arcs. Formally, our drawings map vertices to points in R 2 and edges to curves of one of the following two types.…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

Argyriou¹,

Cornelsen

Förster

et al. 2018

Lecture Notes in Computer Science

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While orthogonal drawings have a long history, smooth orthogonal drawings have been introduced only recently. So far, only planar drawings or drawings with an arbitrary number of crossings per edge have been studied. Recently, a lot of research effort in graph drawing has been directed towards the study of beyond-planar graphs such as 1-planar graphs, which admit a drawing where each edge is crossed at most once. In this paper, we consider graphs with a fixed embedding. For 1-planar graphs, we present algorithms that yield orthogonal drawings with optimal curve complexity and smooth orthogonal drawings with small curve complexity. For the subclass of outer-1-planar graphs, which can be drawn such that all vertices lie on the outer face, we achieve optimal curve complexity for both, orthogonal and smooth orthogonal drawings.

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Section: Introductionmentioning

confidence: 99%

“…Two consecutive segments of an edge meet in a bend. Smooth Orthogonal Layout [5]: Each edge is drawn as a sequence of vertical and horizontal line segments as well as circular arcs: quarter arcs, semicircles, and three-quarter arcs. Consecutive segments must have a common tangent.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

Argyriou¹,

Cornelsen

Förster

et al. 2018

Lecture Notes in Computer Science

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“…Several research directions build upon this successful model. We focus on two models that have recently received attention: (i) the smooth orthogonal [4], in which every edge is a sequence of axis-aligned segments and circular arc segments with common axis-aligned tangents (i.e., quarter, half or three-quarter arc segments), and (ii) the octilinear [2], in which every edge is a sequence of axis-aligned and diagonal (at ±45 • ) segments.…”

Section: Introductionmentioning

confidence: 99%

“…-All planar graphs of max-degree 4 (including the octahedron) admit smooth orthogonal 2-drawings. Note that not all planar graphs of max-degree 4 allow for bendless smooth orthogonal drawings [4], and that such drawings may require exponential area [1]. Bendless smooth orthogonal drawings are possible only for subclasses, e.g., for planar graphs of max-degree 3 [3] and for outerplanar graphs of max-degree 4 [1].…”

Section: Introductionmentioning

confidence: 99%

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

Bekos

Förster

Kaufmann

2018

Lecture Notes in Computer Science

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We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.

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“…Eades [4] proposed a force-directed heuristic method for graph drawing. In recent years, many other methods have been proposed [5,6]. Among these methods, the force-directed method is popular, which uses a heuristic cost or energy function to map the layout of a graph to a real number.…”

Section: Introductionmentioning

confidence: 99%

GPU-Based Parallel Particle Swarm Optimization Methods for Graph Drawing

Liu

Sun³

et al. 2017

Discrete Dynamics in Nature and Society

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Particle Swarm Optimization (PSO) is a population-based stochastic search technique for solving optimization problems, which has been proven to be effective in a wide range of applications. However, the computational efficiency on large-scale problems is still unsatisfactory. A graph drawing is a pictorial representation of the vertices and edges of a graph. Two PSO heuristic procedures, one serial and the other parallel, are developed for undirected graph drawing. Each particle corresponds to a different layout of the graph. The particle fitness is defined based on the concept of the energy in the force-directed method. The serial PSO procedure is executed on a CPU and the parallel PSO procedure is executed on a GPU. Two PSO procedures have different data structures and strategies. The performance of the proposed methods is evaluated through several different graphs. The experimental results show that the two PSO procedures are both as effective as the force-directed method, and the parallel procedure is more advantageous than the serial procedure for larger graphs.

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Smooth Orthogonal Layouts

Cited by 13 publications

References 25 publications

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

GPU-Based Parallel Particle Swarm Optimization Methods for Graph Drawing

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