# Property Testing for Point Sets on the Plane

## Abstract

A *configuration* is a point set on the plane, with no three points collinear. Given three non-collinear points *p*, *q* and \(r\in \mathbb {R}^2\), let \(\chi (p,q,r)\in \{-1,1\}\), with \(\chi (p,q,r)=1\) if and only if, when we traverse the circle defined by those points in the counterclockwise direction, we encounter the points in the cyclic order \(p,q,r,p,q,r,\dots \). For simplicity, extend \(\chi \) by setting \(\chi (p,q,r)=0\) if *p*, *q* and *r* are not pairwise distinct. Two configurations *A* and \(B\subset \mathbb {R}^2\) are said to have the same *order type* if there is a bijection \(\iota :A\rightarrow B\) such that \(\chi (p,q,r)=\chi (\iota (p),\iota (q),\iota (r))\) for all \((p,q,r)\in A^3\). We say that a configuration *C* contains a *copy* of a configuration *A* if there is \(B\subset C\) with *A* and *B* of the same order type. Given a configuration *F*, let \(\mathrm{Forb}(F)\) be the set of configurations that do not contain a copy of *F*. The *distance* between two configurations *A* and *B* with \(|A|=|B|=n\) is given by

where the minimum is taken over all bijections \(\iota :A\rightarrow B\). Roughly speaking, we prove the following property testing result: *being free of a given configuration is efficiently testable*. Our result also holds in the general case of *hereditary properties* \(\mathcal {P}=\mathrm{Forb}(\mathcal {F})\), defined by possibly infinite families \(\mathcal {F}\) of forbidden configurations. Our results complement previous results by Czumaj, Sohler and Ziegler and others, in that we use a different notion of distance between configurations. Our proofs are heavily inspired on recent work of Fox and Wei on testing permutations and also make use of the regularity lemma for semi-algebraic hypergraphs of Fox, Pach and Suk. An extremal function involving order types, which may be of independent interest, plays an important rôle in our arguments.

## Notes

### Acknowledgments

The first author was supported by FAPESP (2014/18641-5). The second author was partially supported by FAPESP (2013/03447-6, 2013/07699-0), CNPq (310974/2013-5, 459335/2014-6) and NUMEC/USP (Project MaCLinC). The third author was supported by CNPq(130483/2016-8). The fourth author was supported by FAPESP (2015/15986-4), CNPq (459335/2014-6) and NUMEC/USP (Project MaCLinC).

## References

- 1.Bland, R.G., Las Vergnas, M.: Orientability of matroids. J. Comb. Theory Ser. B
**24**(1), 94–123 (1978)MathSciNetCrossRefMATHGoogle Scholar - 2.Bokowski, J., Sturmfels, B.: Computational Synthetic Geometry. Lecture Notes in Mathematics. Springer, Berlin (1989). https://doi.org/10.1007/BFb0089253 CrossRefMATHGoogle Scholar
- 3.Chen, X., Freilich, A., Servedio, R.A., Sun, T.: Sample-based high-dimensional convexity testing (June 2017). arXiv:1706.09362
- 4.Czumaj, A., Sohler, C.: Property testing with geometric queries (extended abstract). In: auf der Heide, F.M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 266–277. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44676-1_22 CrossRefGoogle Scholar
- 5.Czumaj, A., Sohler, C., Ziegler, M.: Property testing in computational geometry (extended abstract). In: Paterson, M.S. (ed.) ESA 2000. LNCS, vol. 1879, pp. 155–166. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45253-2_15 CrossRefGoogle Scholar
- 6.Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math.
**2**, 463–470 (1935)MathSciNetMATHGoogle Scholar - 7.Erdős, P., Simonovits, M.: Supersaturated graphs and hypergraphs. Combinatorica
**3**(2), 181–192 (1983). https://doi.org/10.1007/BF02579292 MathSciNetCrossRefMATHGoogle Scholar - 8.Ergün, F., Kannan, S., Kumar, S.R., Rubinfeld, R., Viswanathan, M.: Spot-checkers. J. Comput. Syst. Sci.
**60**(3), 717–751 (2000). https://doi.org/10.1006/jcss.1999.1692. 30th Annual ACM Symposium on Theory of Computing (Dallas, TX 1998)MathSciNetCrossRefMATHGoogle Scholar - 9.Fox, J., Wei, F.: Fast property testing and metrics for permutations (2016). arXiv:161101270
- 10.Fox, J., Pach, J., Suk, A.: A polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing. SIAM J. Comput.
**45**(6), 2199–2223 (2016). https://doi.org/10.1137/15M1007355 MathSciNetCrossRefMATHGoogle Scholar - 11.Goldreich, O. (ed.): Property Testing: Current Research and Surveys. LNCS, vol. 6390. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16367-8 MATHGoogle Scholar
- 12.Goldreich, O., Trevisan, L.: Three theorems regarding testing graph properties. Random Struct. Algorithms
**23**(1), 23–57 (2003). https://doi.org/10.1002/rsa.10078 MathSciNetCrossRefMATHGoogle Scholar - 13.Goodman, J.E., Pollack, R.: Multidimensional sorting. SIAM J. Comput.
**12**(3), 484–507 (1983). https://doi.org/10.1137/0212032 MathSciNetCrossRefMATHGoogle Scholar - 14.Károlyi, G., Solymosi, J.: Erdős-Szekeres theorem with forbidden order types. J. Combin. Theory Ser. A
**113**(3), 455–465 (2006). https://doi.org/10.1016/j.jcta.2005.04.006 MathSciNetCrossRefMATHGoogle Scholar - 15.Richter-Gebert, J., Ziegler, G.M.: Oriented matroids. In: Handbook of Discrete and Computational Geometry, CRC Press Series Discrete Mathematics Application, pp. 111–132. CRC, Boca Raton (1997)Google Scholar
- 16.Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput.
**25**(2), 252–271 (1996)MathSciNetCrossRefMATHGoogle Scholar