Linear space data structures for two types of range search

This paper investigates the existence of linear space data structures for range searching. We examine the homothetic range search problem, where a set S of n points in the plane is to be preprocessed so that for any triangle T with sides parallel to three fixed directions the points of S that lie in T can be computed efficiently. We also look at domination searching in three dimensions. In this problem, S is a set of n points in E³ and the question is to retrieve all points of S that are dominated by some query point. We describe linear space data structures for both problems. The query time is optimal in the first case and nearly optimal in the second.