Placing the Largest Similar Copy of a Convex Polygon Among Polygonal Obstacles
CORNELL UNIV ITHACA NY DEPT OF COMPUTER SCIENCE
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Given a convex polygon P and an environment consisting of polygonal obstacles, we find the largest similar copy of P that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem. Namely, if P is a convex k-gon and if Q has n corners and edges then we can find the placement of the largest similar copy of P in the environment Q in time O k to the 4th power n lambda sub 4 kn log n, where lambda sub 4 is one of the almost-linear functions related to Davenport-Schinzel sequences. If the environment consists only of points then we can find the placement of the largest similar copy of P in time O k-sqn lambda sub 3 kn log n.
- Theoretical Mathematics