Time and space efficient collinearity indexing

The collinearity testing problem is a basic problem in computational geometry, in which, given three sets A, B, C in the plane, of n points each, the task is to detect a collinear triple of points in $A\times B\times C$ or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the collinearity indexing problem, in which we are given two sets A and B, each of n points in the plane, and our goal is to preprocess A and B into a data structure, so that, for any query point $q\in R^2$, we can determine whether q is collinear with a pair of points $(a,b)\in A\times B$. We provide a solution to the problem for the case where the points of A, B lie on an integer grid, and the query points lie on a vertical line, with a data structure of subquadratic storage and sublinear query time. We then extend our result to the case where the query points lie on the graph of a polynomial of constant degree. Our solution is based on the function-inversion technique of Fiat and Naor [11].