# Property Testing in Computational Geometry

@inproceedings{Czumaj2000PropertyTI, title={Property Testing in Computational Geometry}, author={Artur Czumaj and Christian Sohler and Martin Ziegler}, booktitle={Embedded Systems and Applications}, 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.

## 51 Citations

### Testing Geometric Properties ∗ †

- Mathematics, Computer Science
- 2007

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

- Mathematics, Computer ScienceRANDOM-APPROX
- 2003

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

- Computer Science, MathematicsESA
- 2001

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)

- Computer Science
- 2001

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

- Computer Science, MathematicsSoCG
- 2016

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

- Computer Science, MathematicsSODA
- 2014

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

- Mathematics, Computer ScienceSTOC '03
- 2003

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

- Computer ScienceSCG '06
- 2006

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

- Computer Science, MathematicsThe 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
- 2002

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

- Mathematics, Computer ScienceSTOC '06
- 2006

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

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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.

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- Computer ScienceProceedings of 37th Conference on Foundations of Computer Science
- 1996

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.

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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

- Mathematics40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

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

- Mathematics, Computer ScienceSTOC '97
- 1997

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

- Mathematics, Computer ScienceSTOC '98
- 1998

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.

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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.

### Testing Monotonicity

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- 1998

The analysis of the algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it.

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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.

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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.