Efficient Algorithms for Ortho-Radial Graph Drawing
Orthogonal drawings, i.e., embeddings of graphs into grids, are a classic topic in Graph Drawing. Often the goal is to find a drawing that minimizes the number of bends on the edges. A key ingredient for bend minimization algorithms is the existence of an orthogonal representation that allows to describe such drawings purely combinatorially by only listing the angles between the edges around each vertex and the directions of bends on the edges, but neglecting any kind of geometric information such as vertex coordinates or edge lengths.
Barth et al. [2017] have established the existence of an analogous ortho-radial representation for ortho-radial drawings, which are embeddings into an ortho-radial grid, whose gridlines are concentric circles around the origin and straight-line spokes emanating from the origin but excluding the origin itself. While any orthogonal representation admits an orthogonal drawing, it is the circularity of the ortho-radial grid that makes the problem of characterizing valid ortho-radial representations all the more complex and interesting. Barth et al. prove such a characterization. However, the proof is existential and does not provide an efficient algorithm for testing whether a given ortho-radial representation is valid, let alone actually obtaining a drawing from an ortho-radial representation.
In this paper we give quadratic-time algorithms for both of these tasks. They are based on a suitably constrained left-first DFS in planar graphs and several new insights on ortho-radial representations. Our validity check requires quadratic time, and a naive application of it would yield a quartic algorithm for constructing a drawing from a valid ortho-radial representation. Using further structural insights we speed up the drawing algorithm to quadratic running time.
Graph Drawing
Ortho-Radial Graph Drawing
Ortho-Radial Representation
Topology-Shape-Metrics
Efficient Algorithms
Mathematics of computing~Graph algorithms
53:1-53:14
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1903.05048.
Benjamin
Niedermann
Benjamin Niedermann
University of Bonn, Germany
https://orcid.org/0000-0001-6638-7250
Ignaz
Rutter
Ignaz Rutter
University of Passau, Germany
https://orcid.org/0000-0002-3794-4406
Matthias
Wolf
Matthias Wolf
Karlsruhe Institute of Technology, Germany
https://orcid.org/0000-0003-1411-6330
Matthias Wolf was funded by the Helmholtz Program Storage and Cross-linked Infrastructures, Topic 6 Superconductivity, Networks and System Integration.
10.4230/LIPIcs.SoCG.2019.53
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Benjamin Niedermann, Ignaz Rutter, and Matthias Wolf
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