How long will depend on a lot of things. Water quality, lighting and how far the mat has overgrown the current rock (if at all) to name a few. If the star polyp has already overgrown the rock it is attached to then by placing another rock underneath it will allow the unattached leading edge to attach quickly to the new rock. As far as growth... 1-3 weeks for the same 1" piece IME, longer if there is any gap between the polyp rock and the rubble.
SiF
If you remove a strip from the center of the original piece to glue up it will regrow twice as fast as it will be filling in from all directions. Not to mention when it opens up a missing 1/4"-1/2" wide strip is hardly noticable (again if at all)