Michael J. Spriggs

Morphing planar graphs while preserving edge directions

Two straight-line drawings P,Q of a graph (V,E) are called parallel if, for every edge (u,v) ∈ E, the vector from u to v has the same direction in both P and Q. We study problems of the form: given simple, parallel drawings P,Q does there exist a continuous transformation between them such that intermediate drawings of the transformation remain simple and parallel with P (and Q)? We prove that a transformation can always be found in the case of orthogonal drawings; however, when edges are allowed to be in one of three or more slopes the problem becomes NP-hard.

Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree

Abstract Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. For several years, the best upper bound on the time to compute a GMDST was cubic with respect to the number of points in the input set. Recently, Timothy Chan introduced a subcubic time algorithm. In this paper we present an algorithm that generates a tree whose diameter is no more than (1 + ε) times that of a GMDST, for any ε > 0. Our algorithm reduces the problem to several grid-aligned versions of the problem and runs within time

Angles and Lengths in Reconfigurations of Polygons and Polyhedra

O(ε-3+ n) and space O(n).

Cauchy's theorem and edge lengths of convex polyhedra

We explore the possibility of reconfiguring, or “morphing”, one simple polygon to another, maintaining simplicity, and preserving some properties of angles and edge lengths. In linkage reconfiguration, edge lengths must be preserved. By contrast, a monotone morph preserves edge directions and changes edge lengths monotonically. A monotone morph is only possible for parallel polygons—ones with corresponding edges parallel. Our main results are that monotone morphs exist for parallel pairs of polygons that are: (1) convex; or (2) orthogonally convex. Our morphs either move vertices in straight lines, or change few edge lengths at once. On the negative side, we show that it is NP-hard to decide if two simple parallel orthogonal polygons have a monotone morph. We also establish which of these results extend to 3-dimensional polyhedra.

Morphing polyhedra with parallel faces: Counterexamples

In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron--in particular the edge lengths and dihedral angles of the polyhedron. Cauchys rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd.

Parallel morphing of trees and cycles

Two simple polyhedra P and Q (not necessarily convex) are parallel if they share the same edge graph G and each face of P has the same outward-facing unit normal as the corresponding face in Q. Parallel polyhedra P and Q admit a parallel morph if the vertices can be moved in a continuous manner taking us from P to Q such that at all times the intermediate polyhedron determined by the vertex configuration and graph G is both simple and parallel with P (and Q). In this note, we show that even for very restrictive classes of orthogonal polyhedra, there exist parallel polyhedra that do not admit a parallel morph.