Bend-minimum Orthogonal Drawings in Quadratic Time

TL;DR

This paper introduces a quadratic-time algorithm for computing bend-minimum orthogonal drawings of planar 3-graphs, with linear-time solutions under certain constraints, avoiding flow models and limiting bends per edge.

Contribution

It presents the first quadratic-time algorithm for bend-minimum orthogonal drawings in the variable embedding setting for planar 3-graphs, with efficient solutions for constrained cases.

Findings

Algorithm runs in O(n^2) time for general case.

Linear-time algorithm for constrained external face cases.

Drawings have at most two bends per edge.

Abstract

Let $G$ be a planar $3$-graph (i.e., a planar graph with vertex degree at most three) with $n$ vertices. We present the first $O(n^2)$-time algorithm that computes a planar orthogonal drawing of $G$ with the minimum number of bends in the variable embedding setting. If either a distinguished edge or a distinguished vertex of $G$ is constrained to be on the external face, a bend-minimum orthogonal drawing of $G$ that respects this constraint can be computed in $O(n)$ time. Different from previous approaches, our algorithm does not use minimum cost flow models and computes drawings where every edge has at most two bends.