Property Testing in Computational Geometry

@inproceedings{Czumaj2000PropertyTI,
  title={Property Testing in Computational Geometry},
  author={Artur Czumaj and Christian Sohler and Martin Ziegler},
  booktitle={ESA},
  year={2000}
}
We consider the notion of property testing as applied to computational geometry. We aim at developing efficient algorithms which determine whether a given (geometrical) object has a predetermined property Q or is "far" from any object having the property. We show that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the object description size. 

Testing Geometric Properties ∗ †

TLDR
It is shown that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the object description size.

Approximate Testing of Visual Properties

TLDR
This work studies visual properties of discretized images represented by n× n matrices of binary pixel values and obtains algorithms with query complexity independent of n for several basic properties: being a half-plane, connectedness and convexity.

Property Testing with Geometric Queries

TLDR
A number of models are discussed that in the author's opinion fit best geometric problems and apply them to study geometric properties for three very fundamental and representative problems in the area: testing convex position, testing map labeling, and testing clusterability.

Property Testing with Geometric Queries (Extended Abstract)

TLDR
A number of models are discussed that in the author's opinion fit best geometric problems and apply them to study geometric properties for three very fundamental and representative problems in the area: testing convex position, testing map labeling, and testing clusterability.

Testing convexity of figures under the uniform distribution

TLDR
Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing conveXity of figures with 1-sided error.

Testing Surface Area

TLDR
The surface area of an unknown n-dimensional set F given membership oracle access is considered, and the algorithm completely evades the "curse of dimensionality": for any n and any κ > 4/π a 1.27, the "approximation factor" of the testing algorithm.

Sublinear geometric algorithms

We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point

Online geometric reconstruction

TLDR
This work investigates a new class of geometric problems based on the idea of online error correction and provides upper and lower bounds on the complexity of online reconstruction for convexity in 2D and 3D.

Abstract combinatorial programs and efficient property testers

  • A. CzumajC. Sohler
  • Computer Science, Mathematics
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
TLDR
A novel framework for analyzing property testing algorithms with one-sided error is presented and it is shown that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester.

A combinatorial characterization of the testable graph properties: it's all about regularity

TLDR
One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
...

References

SHOWING 1-10 OF 23 REFERENCES

Checking geometric programs or verification of geometric structures

TLDR
This work gives simple and efficient program checkers for some basic geometric tasks and discusses program checking for data structures that have to rely on user-provided functions.

Property testing and its connection to learning and approximation

TLDR
The authors study the question of determining whether an unknown function has a particular property or is /spl epsiv/-far from any function with that property, and devise algorithms to test whether a graph has properties such as being k-colorable or having a /spl rho/-clique.

Handbook of Discrete and Computational Geometry, Second Edition

COMBINATORIAL AND DISCRETE GEOMETRY Finite Point Configurations, J. Pach Packing and Covering, G. Fejes Toth Tilings, D. Schattschneider and M. Senechal Helly-Type Theorems and Geometric

Efficient Testing of Large Graphs

TLDR
This theorem is used to prove that first order graph properties not containing a quantifier alternation of type "/spl forall//spl exist/" are always testable, while it is shown that some properties containing this alternation are not.

Property Testing in Bounded Degree Graphs

TLDR
This work develops the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron and presents randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian.

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

TLDR
The contrapositive statement by which slow convergence implies small cuts in the graph is applied, and this implication is applied in showing that for any graph, the graph vertices can be divided into disjoint subsets such that each subset itself exhibits a certain mixing property that is useful in the analysis.

On the robustness of functional equations

TLDR
It is shown that self-testers and self-correctors can be found for many functions satisfying robust functional equations, including tan x, 1/1+cot x, Ax/1-Ax', cosh x.

Output-sensitive results on convex hulls, extreme points, and related problems

TLDR
Improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints are obtained and an algorithm that computes the vertices of all the convex layers ofP inO(n 2−γ) time for any constant.

Robust Characterizations of Polynomials with Applications to Program Testing

TLDR
The characterizations provide results in the area of coding theory by giving extremely fast and efficient error-detecting schemes for some well-known codes and play a crucial role in subsequent results on the hardness of approximating some NP-optimization problems.

Regular languages are testable with a constant number of queries

TLDR
This paper discusses testability of more complex languages and shows that the query complexity required for testing context free languages cannot be bounded by any function of /spl epsiv/.