Amortized analysis of smooth quadtrees in all dimensions

Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting any given leaf. A quadtree is smooth if any two adjacent leaf boxes differ by at most one in depth. In this paper, we analyze quadtrees that restore smoothness after each split operation, and also maintain neighbor pointers. Our main result shows that the smooth split operation has an amortized cost of at most 2^D⋅(D+1)! auxiliary split operations, which corresponds to amortized constant time in quadtrees of any fixed dimension D. We also show that the exponential dependence on the dimension is unavoidable via a lower bound construction. We additionally give a lower bound construction showing an amortized cost of Ω(log⁡n) for splits in a related quadtree model that does not maintain smoothness.