1975/10/01 by Michael Ian Shamos, Dan Hoey · 11 citations
Computer Science · Business, Management and Accounting · Engineering · Mathematics · #Computational Geometry and Mesh Generation #Facility Location and Emergency Management #Vehicle Routing Optimization Methods #Voronoi diagram #Combinatorics #Upper and lower bounds #Computational geometry #Triangulation #Mathematics #Euclidean space #Plane (geometry) #Delaunay triangulation #Point (geometry) #Euclidean distance #Euclidean geometry #Spanning tree #Set (abstract data type) #Diagram #Line segment #Power diagram #Weighted Voronoi diagram #Tree (set theory) #Algorithm #Computer science #Geometry #Mathematical analysis
paper · doi:10.1109/sfcs.1975.8
openalex publication_date 1975/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.