1986/05/01 by Herbert Edelsbrunner, Leonidas Guibas, Jorge Stolfi · 504 citations
Computer Science · Engineering · Mathematics · #Algorithm #Combinatorics #Computational Geometry and Mesh Generation #Computational geometry #Computer graphics (images) #Computer science #Data structure #Digital Image Processing Techniques #Discrete mathematics #Euclidean geometry #Geometry #Graph #Graphics #Line (geometry) #Line segment #Mathematics #Monotone polygon #Plane (geometry) #Point (geometry) #Point location #Robotics and Sensor-Based Localization #Simple (philosophy) #Subdivision
paper · doi:10.1137/0215023
published in SIAM Journal on Computing 15(2), 317-340 (Society for Industrial and Applied Mathematics)
openalex publication_date 1986/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Point location, often known in graphics as “hit detection,” is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let S denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594–606] for locating a point in S based on separating chains. The new data structure, called a layered dag, can be built in O(m) time, uses O(m) storage, and makes possible point location in O(log m) time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.