I'm looking for a function that can perform something similiar to the Euclidean Allocation process in arcGIS (http://resources.arcgis.com/en/help/main/10.1/index.html#/Euclidean_Allocation/009z0000001m000000/) but using open source tools-- ideally Qgis or Gdal.
Alternatively, if there is a function that can take a polygon layer (e.g. map of the world) and run an algorithm similiar to Thiessen polygons-- that is, fill in the gaps in the extent using the nearest polygon that would be most helpful. For this latter solution, I'd prefer if I didn't have to convert boundaries to points.