Calculating a Unique ID on a Subgroup of Polygons

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I would like to assign a unique ID to polygons in a dataset. The catch to my issue is that I would like polygons that fall within a polygon to have the same unique ID. My example is related to buildings. I have many buildings in a dataset. Each building should have a unique ID, however, buildings that are comprised of multiple polygons (see graphic) should have the same unique ID. I know of a way to do this through a series of dissolves, and spatial joins, but I would like to do this in a calculation. My preference is to do this in ArcMap, but QGIS would also be acceptable.

Any thoughts or help would be appreciated.

Thanks...

EDIT:

Here is a graphic of what I am hoping to resolve. All touching polygons have the same ID. This is unique in the entire dataset, not with the building that is shown.

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