An Output-Sensitive Algorithm for Computing Weighted α-Complexes

An α-complex is a subcomplex of the Delaunay triangulation of a point set P ⊂ R that is topologically equivalent to the union of balls of radius α centered at the points of P . In this paper, we give an output-sensitive algorithm to compute α-complexes of n-point sets in constant dimensions, whose running time is O(f log n log αs ), where s is the smallest pairwise distance and f is the number of simplices in the cα-complex for a constant c. The algorithm is based on a refinement of a recent algorithm for computing the full Delaunay triangulation of P . We also extend the algorithm to work with weighted points provided the weights are appropriately bounded. The new analysis, which may be of independent interest, bounds the number of intersections of k-faces of a Voronoi diagram with (d− k)-faces of the Voronoi diagram of a carefully constructed superset.