Ray shooting and other applications of spanning trees with low stabbing number

Reviewer: Diane Mar Spresser

In this well-written and thorough paper, Agarwal presents several new and interesting applications of spanning trees of low stabbing number, with algorithms that are faster than the best algorithms previously known. The author states that an important goal of the paper is “to demonstrate that such a spanning tree is a versatile tool that can be applied to obtain efficient algorithms for a large class of problems, much beyond the simplex range searching problem for which they were originally introduced” (p. 540). As stated in its abstract, the paper specifically investigates the following problem: “Given a set G of n (possibly intersecting) line segments in the plane, preprocess it so that, given a query ray &rgr; emanating from a point p , one can quickly compute the intersection point F G , r of &rgr; with a line segment of G that lies nearest to p .” As the primary technique, the author uses spanning trees (on the collection of segment endpoints) of low stabbing number. The paper is organized into nine sections. Section 1 summarizes the literature and provides an overview of the remainder of the paper, while Section 2 presents relevant results regarding spanning trees of low stabbing number. In Section 3, the author presents his ray shooting algorithm for arrangements of nonintersecting line segments. Section 4 examines a tradeoff between space and query time: the author shows that the query time for ray shooting in arrange m ents of nonintersecting segments can be improved if more storage is used. Section 5 extends the algorithms for the two previous sections to all intersections between G and a query ray &rgr;. Section 6 then generalizes the ray shooting algorithm to arrangements of possibly intersecting segments. The time and space complexity of the algorithm are analyzed, and a tradeoff between space and query time is derived. Sections 7 and 8 address other problems and applications of the spanning tree technique: implicit point location (Section 7) and polygon containment, implicit hidden surface removal, and polygon placement (Section 8). Section 9 concludes the paper with a summary of significant developments since its submission and a short list of interesting open problems. In general, the paper is extremely well organized and methodical. The writing is clear. The author's care in explaining his work in the context of the work of others is impressive.