On a class of $O(n^2)$ problems in computational geometry

Abstract

There are many problems in computational geometry for which the best know algorithms take time $\Theta (n^2)$ (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 difficult 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 $\Theta (n^2)$ 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 $o(n^2)$ solutions for them unless we can improve the solution for the base problem.