Complexity Analysis of Random Geometric Structures Made Simpler

Olivier Devillers 1 Marc Glisse 1 Xavier Goaoc 2
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Average-case analysis of data-structures or algorithms is com- monly used in computational geometry when the, more clas- sical, worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geo- metric data that can be handled are often simplistic and far from "realistic inputs". We present a new simple scheme for the analysis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed com- plexity analysis. We illustrate our method on two classical structures: con- vex hulls and Delaunay triangulations. Specifically, we give short and elementary proofs of the classical results that n points uniformly distributed in a ball in R^d have a convex hull and a Delaunay triangulation of respective expected d−1 complexities Θ(n^(d+1) ) and Θ(n). We then prove that if we start with n points well-spread on a sphere, e.g. an (epsilon,kappa)-sample of that sphere, and perturb that sample by moving each point randomly and uniformly within distance at most δ of its initial position, then the expected complexity of the convex hull of the resulting point set is Θ( n^(1/2-1/2d) δ^(-d/4+1/4d) ).
Document type :
Conference papers
Liste complète des métadonnées

Cited literature [14 references]  Display  Hide  Download


https://hal.inria.fr/hal-00833774
Contributor : Olivier Devillers <>
Submitted on : Thursday, June 13, 2013 - 2:57:59 PM
Last modification on : Tuesday, December 18, 2018 - 4:18:26 PM
Document(s) archivé(s) le : Saturday, September 14, 2013 - 4:14:03 AM

Files

hal-version.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Olivier Devillers, Marc Glisse, Xavier Goaoc. Complexity Analysis of Random Geometric Structures Made Simpler. 29th Annual Symposium on Computational Geometry, Jun 2013, Rio, Brazil. pp.167-175, ⟨10.1145/2462356.2462362⟩. ⟨hal-00833774⟩

Share

Metrics

Record views

908

Files downloads

222