### Abstract

The order type of a point set in R^{d} maps each (d+1)-tuple of points to its orientation (e.g., clockwise or counterclockwise in R ^{2}). Two point sets X and Y have the same order type if there exists a mapping f from X to Y for which every (d+1 )- Tuple (a_{1}, a _{2},..., a_{d+1}) of X and the corresponding tuple (f(a _{1}), f((a_{2}),..., f(a_{d+1})) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an O (n^{d}) algorithm for this task, thereby improving upon the O(n^{[3d/2]}) algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 405-415 |

Number of pages | 11 |

ISBN (Print) | 9781611973389 |

State | Published - 2014 |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Other

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
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Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 405-415). Association for Computing Machinery.

**The complexity of order type isomorphism.** / Aloupis, Greg; Iacono, John; Langerman, Stefan; Özkan, Özgür; Wuhrer, Stefanie.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014.*Association for Computing Machinery, pp. 405-415, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, OR, United States, 1/5/14.

}

TY - GEN

T1 - The complexity of order type isomorphism

AU - Aloupis, Greg

AU - Iacono, John

AU - Langerman, Stefan

AU - Özkan, Özgür

AU - Wuhrer, Stefanie

PY - 2014

Y1 - 2014

N2 - The order type of a point set in Rd maps each (d+1)-tuple of points to its orientation (e.g., clockwise or counterclockwise in R 2). Two point sets X and Y have the same order type if there exists a mapping f from X to Y for which every (d+1 )- Tuple (a1, a 2,..., ad+1) of X and the corresponding tuple (f(a 1), f((a2),..., f(ad+1)) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an O (nd) algorithm for this task, thereby improving upon the O(n[3d/2]) algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

AB - The order type of a point set in Rd maps each (d+1)-tuple of points to its orientation (e.g., clockwise or counterclockwise in R 2). Two point sets X and Y have the same order type if there exists a mapping f from X to Y for which every (d+1 )- Tuple (a1, a 2,..., ad+1) of X and the corresponding tuple (f(a 1), f((a2),..., f(ad+1)) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an O (nd) algorithm for this task, thereby improving upon the O(n[3d/2]) algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

UR - http://www.scopus.com/inward/record.url?scp=84902106686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902106686&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84902106686

SN - 9781611973389

SP - 405

EP - 415

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

ER -