Multidimensional Data Structures

Abstract

Chapter III was devoted to searching problems in one-dimensional space. In this chapter we will reconsider these problems in higher dimensional space and also treat a number of problems which only become interesting in higher dimensions. Let U be some ordered set and let S ⊆ U^d for some d. An element x ∈ S is a d-tuple (x, …, x_d-1). The simplest searching problem is to specify a point y ∈ U^d and to ask whether y ∈ S; this is called an exact match query and can in principle be solved by the methods of chapter III. Order U^d by lexicographic order and use a balanced search tree. A very general form of query is to specify a region R ⊆ U^d and to ask for all points in R ∩ S. General region queries can only be solved by exhaustive search of set S. Special and more tractable cases are obtained by restricting the query region R to some subclass of regions. Restricting R to polygons gives us polygon searching, restricting it further to rectangles with sides parallel to the axis gives us range searching, and finally restricting the class of rectangles even further gives us partial match retrieval. In one-dimensional space balanced trees solve all these problems efficiently. In higher dimensions we will need different data structures for different types of queries; d-dimensional trees, range trees and polygon trees are therefore treated in VII.2.. There is one other major difference to one-dimensional space. It seems to be very difficult to deal with insertions and deletions; i.e. the data structures described in VII.2. are mainly useful for static sets. No efficient algorithms are known as of today to balance these structures after insertions and deletions. However, there is a general approach to dynamization which we treat in VII.1.. It is applicable to a wide class of problems and yields reasonably efficient dynamic data structures.