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How to short two nodes electronically

I want to make a pcb containing about 32 switches (that do a physical connection between wires) each can be programmed digitally from an arduino board pin. I have a space limitation (as the pcb has to be about the size of a typical mobile phone and contains arduino uno board and some other small components as well) and thus can't use relays to do that. Is there any other way to do it?

P.S. Digital MUX don't serve the purpose since they don't establish a PHYSICAL connection

P.P.S. If it is possible to achieve it at a low cost (<10$) it would be great!