Algorithms for generalized halfspace range searching and other intersection searching problems

In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in R^d, for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in R² for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.