OPTIMAL MORPHS OF PLANAR ORTHOGONAL DRAWINGS

We describe an algorithm that morphs between two planar orthogonal drawings & UGamma;I and & UGamma;O of a graph G, while preserving planarity and orthogonality. Necessarily drawings & UGamma;I and & UGamma;O must be equivalent, that is, there exists a homeomorphism of the plane that transforms & UGamma;I into & UGamma;O. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. (ACM Transactions on Algorithms, 2013). Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of & UGamma;O. We can find corresponding wires in & UGamma;I that share topological properties with the wires in & UGamma;O. The structural difference between the two drawings can be captured by the spirality s of the wires in & UGamma;I, which guides our morph from & UGamma;I to & UGamma;O. We prove that s = O(n) and that s + 1 linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs.