The optimal representation of disjoint iso-oriented rectangles in two-dimensional trees

Abstract

Partitions of the plane are often used to design an efficient data structure for configurations in the plane. In this publication we will use recursive iso-oriented partitions in order to design a search tree data structure for a set S of mutually disjoint iso-oriented rectangles. We will consider two optimization criteria for the choice of the partitions: minimization of the size of the tree and minization of the search time. We will present polynomial time algorithms to find recursive iso-oriented partitions for both optimization criteria separately. We will show that for some arbitrarily large sets S there is a trade-off between the size of the tree and the height of the tree. However, we will prove that for each set S of rectangles logarithmic search time can be obtained with a tree of almost linear size. Such a tree can be found in time almost quadratic in the size of S.