Dynamic multi-dimensional data structures based on quad- and k—d trees

Abstract

Quad-trees and k--d trees have been noted for their lack of dynamic properties as data structures for multi-dimensional point sets. We describe a method to insert points in a quad-tree while keeping the tree balanced that achieves an average time complexity of O(log2 N) per insertion, where N is the number of updates performed on the quad-tree. We define a structure similar to a quad-tree, called a pseudo quad-tree, and show how it can be used to handle both insertions and deletions in O(log2 N) average time. We also discuss how quad-trees and pseudo quadtrees can be extended for use in configurations of points in which more than one point may have a same value in some equal coordinate, without altering the earlier time bounds for insertions, deletions and queries. Similar algorithms are given for k--d trees and the same average time bounds for insertion and deletion are achieved.