Implicitly representing arrangements of lines or segments

An <italic>arrangement</italic> of <italic>n</italic> lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists of <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>) regions, called <italic>faces</italic>. In this paper we study the problem of calculating and storing arrangements <italic>implicitly</italic>, using subquadratic space and preprocessing, so that, given any query point <italic>p</italic>, we can calculate efficiently the face containing <italic>p</italic>. First, we consider the case of lines and show that with Λ(<italic>n</italic>) space<supscrpt>1</supscrpt> and Λ(<italic>n</italic><supscrpt>3/2</supscrpt>) preprocessing time, we can answer face queries in Λ(√<italic>n</italic>) + <italic>&Ogr;</italic>(<italic>K</italic>) time, where <italic>K</italic> is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: 1) given a set of <italic>n</italic> points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, 2) given a simple polygonal path Γ, form a data structure from which we can find the convex hull of any subpath of Γ quickly, and 3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a trade-off between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in Λ(<italic>n</italic><supscrpt>1/3</supscrpt>) time, given Λ(<italic>n</italic><supscrpt>4/3</supscrpt>) space and Λ(<italic>n</italic><supscrpt>5/3</supscrpt>) preprocessing time. Lastly, we note that our techniques allow us to compute <italic>m</italic> faces in an arrangement of <italic>n</italic> lines in time Λ(<italic>m</italic><supscrpt>2/3</supscrpt><italic>n</italic><supscrpt>2/3</supscrpt> + <italic>n</italic>), which is nearly optimal.