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

Summary

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.