Multidimensional Interval Trees

Abstract

The binary interval tree is a random structure that underlies interval division and parking problems. Ageneralization to trees underlying volume partition is investigated in this paper; the size of the associated tree is studied. In d dimensions (arbitrarily high), the momentgenerating function ofthesize of the tree and several pruned variants is shown to satisfy a partial differential equation of order d. So,thesituationis considerably more complex than the linear case that arises in one dimension. The paper addresses volume partitionby points, andthe case canbesolved,in contrast with a well-known parking problem,where the partition is done by solid objects; the parking analog remains unsolved, but avariation is shown to be tractable,which may glean research toward an eventual solution to the standard parking problem may. The multidimensional interval tree can be viewed as acontinuous analog of the discrete quadtrees.