Analysis of the n-dimensional quadtree decomposition for arbitrary hyperrectangles

We give a closed-form expression for the average number of n-dimensional quadtree nodes ("pieces" or "blocks") required by an n-dimensional hyperrectangle aligned with the axes. Our formula includes as special cases the formulae of previous efforts for two-dimensional spaces. It also agrees with theoretical and empirical results that the number of blocks depends on the hypersurface of the hyperrectangle and not on its hypervolume. 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 two-dimensional spaces (geographic information systems and spatial databases in general), as well in higher dimensionality spaces (as oct-trees for three-dimensional spaces, e.g., in graphics, robotics, and three-dimensional medical images). Our formula permits the estimation of the space requirements for data hyperrectangles when stored in an index structure like a (n-dimensional) quadtree, as well as the estimation of the search time for query hyperrectangles, for the so-called linear quadtrees. 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 hyperrectangle.