An orthogonal drawing is an embedding of a plane graph into a grid. In a seminal work of Tamassia (SIAM Journal on Computing 1987), a simple combinatorial characterization of angle assignments that can be realized as bend-free orthogonal drawings was established, thereby allowing an orthogonal drawing to be described combinatorially by listing the angles of all corners. The characterization reduces the need to consider certain geometric aspects, such as edge lengths and vertex coordinates, and simplifies the task of graph drawing algorithm design. Barth, Niedermann, Rutter, and Wolf (SoCG 2017) established an analogous combinatorial characterization for ortho-radial drawings, which are a generalization of orthogonal drawings to cylindrical grids. The proof of the characterization is existential and does not result in an efficient algorithm. Niedermann, Rutter, and Wolf (SoCG 2019) later addressed this issue by developing quadratic-time algorithms for both testing the realizability of a given angle assignment as an ortho-radial drawing without bends and constructing such a drawing. In this paper, we further improve the time complexity of these tasks to near-linear time. We establish a new characterization for ortho-radial drawings based on the concept of a good sequence. Using the new characterization, we design a simple greedy algorithm for constructing ortho-radial drawings.

翻译：正交绘图是将平面图嵌入到网格中的一种表示。在Tamassia开创性工作（SIAM Journal on Computing, 1987）中，建立了可实现为无折线正交绘图的角分配的简单组合刻画，从而允许通过列出所有拐点的角度以组合方式描述正交绘图。该刻画降低了对边长度和顶点坐标等几何因素的考量需求，简化了图绘制算法的设计任务。Barth、Niedermann、Rutter和Wolf（SoCG, 2017）为基于圆柱网格的正交绘图推广形式——正交径向绘图建立了类似的组合刻画。该刻画的证明是存在性的，未能导出高效算法。Niedermann、Rutter和Wolf（SoCG, 2019）后续通过开发二次时间算法解决了该问题，该算法既可检验给定角分配能否实现为无折线正交径向绘图，又能构造此类绘图。本文进一步将这些任务的时间复杂度改进至近乎线性时间。我们基于“好序列”概念建立了正交径向绘图的新刻画，并利用该新刻画设计了构造正交径向绘图的简单贪心算法。