Computer algebra: symbolic and algebraic computation (2nd ed.)
Fast algorithms for the characteristic polynomial
Theoretical Computer Science
Computational geometry: an introduction
Computational geometry: an introduction
Computing a ham-sandwich cut in two dimensions
Journal of Symbolic Computation
Topologically sweeping an arrangement
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A geometric consistency theorem for a symbolic perturbation scheme
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
A general approach to removing degeneracies
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Computing Roadmaps of General Semi-Algebraic Sets
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A practical method for the sparse resultant
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
On geometric optimization with few violated constraints
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Interactive visualization of weighted three-dimensional alpha hulls
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
ACM Computing Surveys (CSUR)
Polyhedral modelling with exact arithmetic
SMA '95 Proceedings of the third ACM symposium on Solid modeling and applications
Output-sensitive results on convex hulls, extreme points, and related problems
Proceedings of the eleventh annual symposium on Computational geometry
Controlled simplification of genus for polygonal models
VIS '97 Proceedings of the 8th conference on Visualization '97
Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
On degeneracy in geometric computations
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Topology Simplification for Polygonal Virtual Environments
IEEE Transactions on Visualization and Computer Graphics
Computing Smooth Molecular Surfaces
IEEE Computer Graphics and Applications
Fast analytical computation of Richards's smooth molecular surface
VIS '93 Proceedings of the 4th conference on Visualization '93
SIAM Journal on Optimization
| Hi-index | 0.00 |
Our aim is to perturb the input so that an algorithm designed under the hypothesis of input non-degeneracy can execute on arbitrary instances. The deterministic scheme of [EmCa] was the first efficient method and was applied to two important predicates. Here it is extended in a consistent manner to another two common predicates, thus making it valid for most algorithms in computational geometry. It is shown that this scheme incurs no extra algebraic complexity over the original algorithm while it increases the bit complexity by a factor roughly proportional to the dimension of the geometric space. The second contribution of this paper is a variant scheme for a restricted class of algorithms that is asymptotically optimal with respect to the algebraic as well as the bit complexity. Both methods are simple to implement and require no symbolic computation. They also conform to certain criteria ensuring that the solution to the original input can be restored from the output on the perturbed input. This is immediate when the input to solution mapping obeys a continuity property and requires some case-specific work otherwise. Finally we discuss extensions and limitations to our approach.