Randomized Insertion and Deletion in Point Quad Trees

Abstract

In this work we introduce randomized insertion and deletion algorithms for quad trees. These algorithms are simple and they are defined for any dimension K (where K is a natural number), in contrast with the fact that standard deletion in quad trees is rather complicated and becomes more cumbersome as K increases [12]. In addition, in online applications, where long sequences of random interleaved insertions and deletions are performed, quad trees built and dynamically maintained by the randomized algorithms presented here (randomized quad trees) preserve their expected path length, a measure that augments when, in similar circumstances, the standard deletion algorithm is used [4]. Moreover, the expected values of random variables (such as internal path length, depth, cost of successful or unsuccessful search, cost of partial match queries, between others) given in the literature for random quad trees are valid for the randomized quad trees introduced in this work.

This research was partially supported by the Future and Emergent Technologies programme of the EU under contract IST-1999-14186 (ALCOM-FT) and the Spanish Min. of Science and Technology project TIC2002-00190 (AEDRI II).