Sublinear Geometric Algorithms

We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in two-dimensional triangulations and Voronoi diagrams, and ray shooting in convex polyhedra, all of which run in expected time $O(\sqrt{n}\,)$, where $n$ is the size of the input. We also provide sublinear solutions for the approximate evaluation of the volume of a convex polytope and the length of the shortest path between two points on the boundary.