Fill larger polygon empty space with closest subset of polygons

Looking for an ArcGIS or QGIS solution.

I need to divide up the space of the large single green polygon. Each new polygon would represent "nearness" to the purple polygons and have the ID of the purple Polygon it is closest to. Need to fill all the empty green space.

This would be a Thiessen Polygons solution if the purple were point but my input is a polygon and I need the equivalent but "what polygon am I closest to" and the ID of the original polygon has to carry over to the new closest polygons. I am open to turning the purple polygons into raster and having a raster solution.

I know I can turn the polygon boundary into equally spaced point and do this but I am looking for something more elegant and fully accurate.