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.

翻译：本文研究了计算具有最小曲线复杂性的双向平面测谎半上平面图案的问题,即绘制能够最大限度地减少每边缘弯曲的最大数目的图案。 我们证明,每双向平面测谎都承认具有曲线复杂性2的准上向平面图案,这是最坏的情况。 我们还表明,在准上向平面测图中将曲线复杂性最小化的问题可以模拟成单位-能力平面图流网络的微成本流问题。 这就产生了一个计算出具有最低曲线复杂性的准上向平面图案的时间算法,此外,在没有边缘弯曲超过两倍的情况下,该图案的弯曲最小值是双向平面图案的两倍。 相比之下,我们展示了双向平面测图案,其弯曲-最小度准上下方平面图案需要直线曲线复杂性,即使在可变嵌入的设置中也是如此。