Efficient searching with linear constraints

We show how to preprocess a set S of points in R^d into an external memory data structure that efficiently supports linear-constraint queries. Each query is in the form of a linear constraint a·x≤b; the data structure must report all the points of S that satisfy the constraint. Our goal is to minimize the number of disk blocks required to store the data structure and the number of disk accesses (I/Os) required to answer a query. For d = 2, we present the first near-linear size data structure that can answer linear-constraint queries using an optimal number of I/Os. We also present a linear-size data structure that can answer queries efficiently in the worst case. We combine these two approaches to obtain tradeoffs between space and query time. Finally, we show that some of our techniques extend to higher dimensions.