The alpha complex is a fundamental data structure from computational geometry, which encodes the topological type of a union of balls $B(x; r) \subset \mathbb{R}^m$ for $x\in S$, including a weighted version that allows for varying radii. It consists of the collection of "simplices" $\sigma = \{x_0, ..., x_k \} \subset S$, which correspond to nomempty $(k + 1)$-fold intersections of cells in a radius-restricted version of the Voronoi diagram. Existing algorithms for computing the alpha complex require that the points reside in low dimension because they begin by computing the entire Delaunay complex, which rapidly becomes intractable, even when the alpha complex is of a reasonable size. This paper presents a method for computing the alpha complex without computing the full Delaunay triangulation by applying Lagrangian duality, specifically an algorithm based on dual quadratic programming that seeks to rule simplices out rather than ruling them in.

翻译：Alpha复形是计算几何中的一种基本数据结构，它编码了球体集合$B(x; r) \subset \mathbb{R}^m$（其中$x \in S$）的并集的拓扑类型，包括允许半径变化的加权版本。该复形由"单纯形"$\sigma = \{x_0, ..., x_k \} \subset S$的集合构成，这些单纯形对应Voronoi图在半径受限版本中细胞的非空$(k+1)$重交集。现有计算alpha复形的算法要求点位于低维空间中，因为它们首先计算完整的Delaunay复形，而即使alpha复形规模合理，这一过程也会迅速变得难以处理。本文提出一种无需计算完整Delaunay三角剖分即可计算alpha复形的方法，通过应用拉格朗日对偶性，具体而言是一种基于对偶二次规划的算法，该算法倾向于排除单纯形而非纳入它们。