Computational geometry: an introduction
Computational geometry: an introduction
Worst-case optimal hidden-surface removal
ACM Transactions on Graphics (TOG)
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
The complexity and construction of many faces in arrangements of lines and of segments
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Generalized hidden surface removal
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
On the union of fat wedges and separating a collection of segments by a line
Computational Geometry: Theory and Applications
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
The complexity of the free space for a robot moving amidst fat obstacles
Computational Geometry: Theory and Applications
How hard are n2-hard problems?
ACM SIGACT News
ACM SIGACT News
Robot Motion Planning
The Visibility Diagram: a Data Structure for Visibility Problems and Motion Planning
SWAT '90 Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Better lower bounds on detecting affine and spherical degeneracies
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
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There are many problems in computational geometry for which the best know algorithms take time @Q(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 @Q(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.