Morphing Planar Graph Drawings via Orthogonal Box Drawings

Therese Biedl, Anna Lubiw, Jack Spalding-Jamieson
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Abstract

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph on an $O(n) \times O(n)$ grid. The planarity-preserving morph is composed of $O(n)$ linear morphs between successive pairs of drawings, each on an $O(n) \times O(n)$ grid with a constant number of bends per edge. The algorithm to compute the morph runs in $O(n^2)$ time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.
通过正交框图变形平面图形绘制
我们给出了一种变形平面图绘制的算法,这种算法以允许每条边有一定数量的弯曲为代价,实现了较小的网格大小。输入是一个 $n$ 有顶点的平面图和在 $O(n) \times O(n)$ 网格上的两个平面直线图。保持平面性的变形由连续两幅图之间的 $O(n)$ 线性变换组成,每幅图在 $O(n) 次 O(n)$ 网格上,每条边的弯曲次数恒定。计算变形的算法在字 RAM 模型上以 $O(n^2)$ 的时间运行,不需要算术运算,特别是不需要平方根或立方根。算法的第一步是将每个输入图形变形为平面正交框图,其中顶点用框表示,每条边绘制为水平或垂直线段。第二步是在平面正交方块图之间进行变形。这是通过扩展已知的技术,将平面正交绘图与点顶点变形来实现的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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