Analysis of the n-dimensional quadtree decomposition for arbitrary hyper-rectangles

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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 efforts for 2-dimensional spaces \cite{Faloutsos92Analytical}. 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 [Arya et al., 1994]. 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. (Also cross-referenced as UMIACS-TR-94-130)