On local transformation of polygons with visibility properties

Abstract

One strategy for the enumeration of a class of objects is local transformation, in which new objects of the class are produced by means of a small modification of a previously-visited object in the same class. When local transformation is possible, the operation can be used to generate objects of the class via random walks, and as the basis for such optimization heuristics as simulated annealing. For general simple polygons on fixed point sets, it is still not known whether the class of polygons on the set is connected via a constant-size local transformation. In this paper, we exhibit a simple local transformation for which the following polygon classes are connected: monotone, x-monotone, star-shaped, (weakly) edge-visible and (weakly) externally visible. The latter class is particularly interesting as it is the most general polygon class known to be connected under local transformation. For each of the polygon classes, we also provide asymptotically-tight worst-case upper bounds on the minimum number of operations required to transform one member of the class to any other. © 2002 Elsevier Science B.V. All rights reserved.