Greg Aloupis, John Iacono, Stefan Langerman, Özgür Özkan, Stefanie Wuhrer
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引用次数: 0
摘要
在 \(\mathbb {R}^d\)中一个点集的秩类型映射每个 \((d{+}1)\)-tuple of points 到它的方向(例如,在 \(\mathbb {R}^2\)中顺时针或逆时针)。如果存在从 X 到 Y 的双投射 f,且 X 中的每\((d{+}1)\)-元组\((a_1,a_2,\ldots ,a_{d+1})\)和 Y 中的相应元组\(f(a_1),f(a_2),\ldots ,f(a_{d+1}))具有相同的方向,则两个点集 X 和 Y 具有相同的阶类型。本文研究了判断两个点集是否具有相同阶类型的复杂性。我们为这个任务提供了一个(O(n^d))算法,从而改进了 Goodman 和 Pollack 的(O(n^{\lfloor {3d/2}\rfloor })算法(SIAM J. Comput.12(3):484-507, 1983).该算法只使用阶类型查询,也适用于抽象阶类型(或非循环定向矩阵)。如果算法只使用阶类型查询,那么我们的算法无论是在抽象环境中还是对于可实现的点集都是最优的。
The order type of a point set in \(\mathbb {R}^d\) maps each \((d{+}1)\)-tuple of points to its orientation (e.g., clockwise or counterclockwise in \(\mathbb {R}^2\)). Two point sets X and Y have the same order type if there exists a bijection f from X to Y for which every \((d{+}1)\)-tuple \((a_1,a_2,\ldots ,a_{d+1})\) of X and the corresponding tuple \((f(a_1),f(a_2),\ldots ,f(a_{d+1}))\) 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 \(O(n^{\lfloor {3d/2}\rfloor })\) algorithm of Goodman and Pollack (SIAM J. Comput. 12(3):484–507, 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.
期刊介绍:
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.