Analysis of the N-dimensional Quadtree Decomposition for Arbitrary Hyper-rectangles

We give a closed-form expression for the average number of n-dimensional quadtree nodes (`pieces' or`blocks') required by an n-dimensional hyper-rectangle aligned with the axes. Our formula includes as special cases the formulae of previous eeorts for 2-dimensional spaces 8]. It also agrees with theoretical and empirical results that the number of blocks depends on the hyper-surface of the hyper-rectangle and not on its hyper-volume. The practical use of the derived formula is that it allows the estimation of the space requirements of the n-dimensional quadtree decomposition. Quadtrees are used extensively in 2-dimensional spaces (geographic information systems and spatial databases in general), as well in higher dimensionality spaces (as oct-trees for 3-dimensional spaces, e.g. in graphics, robotics and 3-dimensional medical images 2]). Our formula permits the estimation of the space requirements for data hyper-rectangles when stored in an index structure like a (n-dimensional) quadtree, as well as the estimation of the search time for query hyper-rectangles. A theoretical contribution of the paper is the observation that the number of blocks is a piece-wise linear function of the sides of the hyper-rectangle.