Ortho-Radial Drawing in Near-Linear Time

TL;DR

The paper addresses the problem of efficiently determining and constructing ortho-radial drawings from representations. It introduces a new good-sequence framework and a greedy algorithm that, for a fixed reference edge, runs in $O(n\log n)$ time (or $O(n\log^2 n)$ without a fixed reference), delivering a drawing when drawable or certifying non-drawability via a strictly monotone cycle. It leverages a sequence-based decomposition of horizontal segments and a careful preprocessing/augmentation strategy to avoid costly left-first DFS traversals typical of prior approaches. The results substantially improve previous quadratic-time methods, offering near-linear-time performance and paving the way for practical application and further theoretical exploration of the topology-shape-metric framework in ortho-radial drawing. The work also establishes a reduction to biconnected simple graphs to generalize the approach while preserving drawability properties, and discusses open questions in topology-to-shape translation and bend minimization for ortho-radial drawings.

Abstract

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.

Figures (19)

• Figure 4: A grid, an orthogonal drawing, a cylindrical grid, and an ortho-radial drawing.
• Figure 5: A non-crossing-free path, a crossing-free path, and a facial cycle.
• Figure 6: A drawing of an ortho-radial representation with a reference edge, where the small blue circles in the left figure denote the angles in the representation that are realized in the right figure.
• Figure 7: The calculation of $\mathsf{direction}(e^\star, P, e)$.
• Figure 8: Changing the reference edge to $e$ leads to a strictly monotone cycle.

Theorems & Definitions (25)

• Lemma 2.1: barth2023topology
• Lemma 2.2: barth2023topology
• Lemma 2.3: barth2023topology
• Lemma 2.4: barth2023topology
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 5.1