So a couple of question to pose first:
- Can you live at home? If so, do you plan on living at home?
- Are you looking for scholarships?
- Are you working while at school?
- Why are you going for absolute cheapest? Why not best value?
I ask these because they are important questions that will shape what you look for.
I hesitate to say "cheapest" because like many other things, you will get what you pay for. If you go to a crappy college that does not have any prospects of getting you a EE job, then you just wasted a lot of money. I think best value is looking into a 2-year community college that will transfer to a state school in state. Make sure the in state school is a decent engineering school, and it has partnerships with engineering companies. Those schools are generally "feeders" into companies, and will offer internships and jobs after college.
Housing is not a negligible cost either. If living at home is an option (understandably, it may not be), that will also save a lot of money as well. If that is not an option, you will likely have to work while in school to keep costs down. I personally worked as a Resident Assistant and I got free room, board, and a food allowance. But you can also find paid internships during the year that are full time in the summer that will defer costs. My brother did that, and was offered a full time job at a major company afterwards.
On the other hand, there are scholarships as well. I know that many government jobs offer full ride scholarships in exchange for working for then for x years afterwards (usually 4 years). ROTC will not only give you a scholarship, but will offer subsidies as well.