On the size of quadtrees generalized to d-dimensional binary pictures

AbstractSome results about the size of quadtrees and linear quadtrees, used to represent binary 2n × 2n digital pictures, are generalized to d-dimensional 2n × … × 2n pictures. Among these results are a comparison of the space-efficiency of linear vs regular trees, in terms of both the number of nodes of the tree and the number of bits needed to store each node, and an upper bound on the number of nodes as a function of n and the perimeter of the picture

Using topological sweep to extract the boundaries of regions in maps represented by region quadtrees

A variant of the plane sweep paradigm known as topological sweep is adapted to solve geometric problems involving two-dimensional regions when the underlying representation is a region quadtree. The utility of this technique is illustrated by showing how it can be used to extract the boundaries of a map in O(M) space and O(Ma(M)) time, where M is the number of quad tree blocks in the map, and a(·) is the (extremely slowly growing) inverse of Ackerman's function. The algorithm works for maps that contain multiple regions as well as holes. The algorithm makes use of active objects (in the form of regions) and an active border. It keeps track of the current position in the active border so that at each step no search is necessary. The algorithm represents a considerable improvement over a previous approach whose worst-case execution time is proportional to the product of the number of blocks in the map and the resolution of the quad tree (i.e., the maximum level of decomposition). The algorithm works for many different quadtree representations including those where the quadtree is stored in external storage