# Finding a way out of a polygon

There is a simply-connected polygon $C$. It contains $n$ pairwise-interior-disjoint simply-connected polygons, $D_1,\dots,D_n$:

The goal is to select one of the polygons, say $D_i$, and attach to it a polygon $E_i$ such that:

• $D_i\cup E_i$ is still simply-connected.
• $E_i$ is interior-disjoint from all other polygons $D_i$.
• $D_i\cup E_i$ is connected with the exterior of $C$:

Is there an algorithm, preferably published, that solves this problem in time polynomial in the representation of the problem (e.g. total number of vertices of all polygons)?

Basically this is shortest path problem in polygon with holes. There is no constraint in taking $E_i$ as a path of negligible width.