Linear-time border-tracing algorithms for quadtrees

In applications where the quadtree is used as an underlying object representation, a number of basic operations are implemented as a trace along the border of the object's region. A technique is presented that determines a way to shift any given scene (as well as its quadtree), so that the border of all the objects in the scene can be traversed in time proportional to the length of all the borders in the scene (or the number of blocks when the scene is represented as a quadtree). This determination is shown to be performed in time proportional to the length of all the borders in the scene. This allows the direct translation of a number of chain-code algorithms into quadtree algorithms without loss of asymptotic worst-case efficiency. This results in improved worst-case analyses of algorithms that convert chain codes into quadtrees and that perform connected component labeling of images represented as quadtrees.