Placing the Largest Similar Copy of a Convex Polygon among Polygonal Obstacles

摘要

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-sq)n lambda sub 3 (kn) log n). (jhd)