[OT] Text stream editing problem

[csj <csj@zapo.net>]

Let's say I want to write a script that can convert the
following:

A-0001     Directory_1/Subdirectory_1/File_1.txt
A-0002     ./Directory_2/Subdirectory_2/File_2.txt
A-0003     ./Directory_3/Subdirectory_3/Subdirectory_4/File_3.txt

to:

A-0001     /Subdirectory_1/File_1.txt
A-0002     /Subdirectory_2/File_2.txt
A-0003     /Subdirectory_3/Subdirectory_4/File_3.txt

My basic strategy would be to sed 's|/Directory_.||g'.
Unfortunately this isn't of universal application.  I'm looking
for a solution than can take into account all possible names for
"Dir_Foo".

The solution I have in mind is to be able to count the "/" and
then cut, say after the Nth match (I will of course have to put a
"./" before every path that doesn't begin with one).  Is there a
quick way to do this using only the GNU coreutils?