Considering the political and historical context in which the term perestroika is situated, I feel it deserves more than a simple translation to english. So for your edification and wonderment I would like to now provide a complex translation, with a little bit of etymology thrown in to spice things up.
Perestroika: is a noun formed from the verb perestroit', with two meaningful parts: the prefix pere- and the verb stroit'. Stroit' means "to build/construct". The prefix pere- carries a lot of meanings, a few relevant here are: indicating movement over or across something; repetition of an action; movement from one place to another; redirection; movement involving lots of people or objects; indicating intermediate stage of action; indicating superiority in something.
So the basic meaning of perestroit' is "to rebuild", and the noun perestroika formed from it means "rebuilding, or reconstruction" but you can see that the prefix pere- carries a lot more nuance to the term than what is indicated in English by the term "rebuilding". Here reconstruction is a more accurate translation; because of its use in context with the Reconstruction of the South after the Civil War, it begins to carry some of the weight present in the Russian.