Input-Sensitive Compliant Motion in the Plane

In 1989, Briggs [1] gave an O(n2 log n) algorithm for single-step planar compliant motion planning among polygonal obstacles. At the same time, Friedman, Hershberger, and Snoeyink [8] gave an O(n log n) preprocessing, O(n) space, O(log n) query time algorithm for the same problem when the environment is the interior of a simple polygon. In this paper, we bridge the complexity gap between these two results. For a polygonal environment with n vertices and k connected components on its boundary, our algorithm requires O(kn log n) preprocessing time, O(kn log n) space (the preprocessing phase requires O(kn log n) space to run, and produces a data structure of size O(kn)), and O(k log n) query time.