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：正交图绘制是将平面图嵌入网格的过程。在Tamassia的开创性工作（《SIAM计算杂志》1987年）中，建立了角度分配可实现为无弯曲正交图绘制的简单组合特征，从而允许通过列出所有转角的角度来组合描述正交图绘制。该特征减少了对某些几何方面（如边长和顶点坐标）的考虑，并简化了图绘制算法设计的任务。Barth、Niedermann、Rutter和Wolf（SoCG 2017）为正交径向图建立了类似的组合特征，这是正交图向柱面网格的推广。该特征的证明是存在性的，并未产生高效算法。Niedermann、Rutter和Wolf（SoCG 2019）后来通过开发二次时间算法解决了这一问题，该算法既能测试给定角度分配作为无弯曲正交径向图的可实现性，又能构建此类图。在本文中，我们将这些任务的时间复杂度进一步改进至近线性时间。基于良序概念，我们为正交径向图建立了新的特征。利用新特征，我们设计了一种简单的贪心算法来构建正交径向图。