Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
J. Pach , City Coll., CUNY, NY, USA
G. Tardos , City Coll., CUNY, NY, USA
A triangle is said to be /spl delta/-fat if its smallest angle is at least /spl delta/<0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of /spl delta/-far triangles in the plane determines at most O (n//spl delta/ log 2//spl delta/) holes. This improves on some earlier bounds of (Efrat et al., 1993; Matousek et al., 1994). Solving a problem of (Agarwal and Bern, 1999) we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.
computational complexity; computational geometry; boundary complexity; fat triangle union; holes; upper bounds; motion planning; separator line; computational geometry
G. Tardos and J. Pach, "On the boundary complexity of the union of fat triangles," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 423.