Quasi-upward Planar Drawings with Minimum Curve Complexity

TL;DR

This paper investigates how to create quasi-upward planar drawings of bimodal plane digraphs with the least number of bends per edge, providing optimal bounds and an efficient algorithm for minimizing curve complexity.

Contribution

It proves that every bimodal plane digraph admits a quasi-upward drawing with at most two bends per edge and models the problem as a min-cost flow for efficient computation.

Findings

Every bimodal plane digraph has a quasi-upward drawing with at most two bends per edge.

An $ ilde{O}(m^{4/3})$-time algorithm computes minimum curve complexity drawings.

Some bimodal planar digraphs require linear curve complexity even with variable embeddings.

Abstract

This paper studies the problem of computing quasi-upward planar drawings of bimodal plane digraphs with minimum curve complexity, i.e., drawings such that the maximum number of bends per edge is minimized. We prove that every bimodal plane digraph admits a quasi-upward planar drawing with curve complexity two, which is worst-case optimal. We also show that the problem of minimizing the curve complexity in a quasi-upward planar drawing can be modeled as a min-cost flow problem on a unit-capacity planar flow network. This gives rise to an $\tilde{O}(m^\frac{4}{3})$-time algorithm that computes a quasi-upward planar drawing with minimum curve complexity; in addition, the drawing has the minimum number of bends when no edge can be bent more than twice. For a contrast, we show bimodal planar digraphs whose bend-minimum quasi-upward planar drawings require linear curve complexity even in the variable embedding setting.