An exact closed-form formula for d-dimensional quadtree decomposition of arbitrary hyperrectangles

In this paper, we solve the classic problem of computing the average number of decomposed quadtree blocks (quadrants, nodes, or pieces) in quadtree decomposition for an arbitrary hyperrectangle aligned with the axes. We derive a closed-form formula for general cases. The previously known state-of-the-art solution provided a closed-form solution for special cases and utilized these formulas to derive linearly interpolated formulas for general cases individually. However, there is no exact and uniform closed-form formula that fits all cases. Contrary to the top-down counting approach used by most prior solutions, we employ a bottom-up enumeration approach to transform the problem into one that involves the Cartesian product of d multisets of successive 2's powers. Classic combinatorial enumeration techniques are applied to obtain an exact and uniform closed-form formula. The result is of theoretical interest since it is the first exact closed-form formula for arbitrary cases. Practically, it is nice to have a uniform formula for estimating the average number since a simple program can be conveniently constructed taking side lengths as inputs. © 2006 IEEE.