A Functional Approach to Data Structures and Its Use in Multidimensional Searching

We establish new upper bounds on the complexity of multidimensional 3earching. Our results include, in particular, linear-size data structures for range and rectangle counting in two dimensions with logarithmic query time. More generally, we give improved data structures for rectangle problems in any dimension, in a static as well as a dynamic setting. Several of the algorithms we give are simple to implement and might be the solutions of choice in practice. Central to this paper is the nonstandard approach followed to achieve these results. At its root we find a redefinition of data structures in terms of functional specifications. 1. Introduction. This paper has two main parts: in 2, we discuss a method for transforming data structures using functional specifications; in the remaining sections, we use such transformations to solve a number of problems in multidimensional searching. To begin with, let us summarize the complexity results of this paper. The generalization of the notion of rectangle in higher dimensions is called a d-range: it is defined as the Cartesian product of d closed intervals over the reals. Let