On a class of O(n²) problems in computational geometry

Abstract

There are many problems in computational geometry for which the best know algorithms take time (n2) (or more) in the worst case while only very low lower bounds are known. In this paper we describe a large class of problems for which we prove that they are all at least as dicult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0. We call such problems 3sum-hard. The best known algorithm for the base problem takes (n2) time. The class of 3sum-hard problems includes problems like: Given a set of lines in the plane, are there three that meet in a point?; or: Given a set of triangles in the plane, does their union have a hole? Also certain visibility and motion planning problems are shown to be in the class. Although this does not prove a lower bound for these problems, there is no hope of obtaining o(n2) solutions for them unless we can improve the solution for the base problem