Embedding Quadtrees into Hypercubes

Abstract

The quadtree is useful data structure for a variety of image processing. In this paper, we propose two simple but effective methods for embedding quadtrees into hypercubes. First, we embed a complete quadtree of height h into a (3(h-1)+4)-dimensional hypercube, or into a smaller incomplete hypercube which comprises a (3(h-1)+3)-dimensional hypercube and a (3(h-2)+4)-dimensional hypercube. This embedding preserves the adjacency of the complete quadtree, while the second method does not. The second method is to embed a complete quadtree of height h into an incomplete hypercube of the same node size with the congestion 2 and the dilation is at most 3.