Efficient Searching with Linear Constraints

P. K. Agarwal, L. A. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter. ``Efficient Searching with Linear Constraints,'' Journal of Computer and System Sciences, 61(2), October 2000, 194-216. An extended abstract appears in Proceedings of the 17th Annual ACM Symposium on Principles of Database Systems (PODS '98), Seattle, WA, June 1998, 169-178.

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We show how to preprocess a set S of points in d-dimensional Euclidean space to get an external memory data structure that efficiently supports linear-constraint queries. Each query is in the form of a linear constraint $\mathbf{a \cdot x \leq b}$; the data structure must report all the points of S that satisfy the query. (This problem is called halfspace range searching in the computational geometry literature.) 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 structures 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.