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.

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U2 - 10.1137/1.9781611973402.30

DO - 10.1137/1.9781611973402.30

M3 - Conference contribution

AN - SCOPUS:84902106686

SN - 9781611973389

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 405

EP - 415

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

PB - Association for Computing Machinery

T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

Y2 - 5 January 2014 through 7 January 2014

ER -