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 thehomothetic range search problem, where a setS ofn points in the plane is to be preprocessed so that for any triangleT with sides parallel to three fixed directions the points ofS that lie inT can be computed efficiently. We also look atdomination searching in three dimensions. In this problem,S is a set ofn points inE ³ and the question is to retrieve all points ofS 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.This research was conducted while the first author was with Brown University and the second author was with the Technical University of Graz, Austria. The first author was supported in part by NSF Grant MCS 83-03925.