The 01-swap operation, b, on a binary string S, permutes all occurences of 01 within the original string to 10 (the process is not recursive). For instance:

b(0) = 0, b(1) = 1, b(00) = 00, b(01) = 10, b(11) = 11

b(000) = 000, b(001) = 010, b(010) = 100, b(100) = 100

b(0 01 01 1 01 0) = 0 10 10 1 10 0 (spacing added for clarity).

Design a circuit using 2-1 multiplexers that can be used to perform the 01-swap on a k-bit string S = Sk1Sk2 S0. where each Si is a bit.

(a) First, show the circuit, built using the 2:1 MUX, whose output is the ith bit of b(S) where 0 < i < k 1. YOU DO NOT NEED ANY AND, OR, OR NOT GATES, only a single 2:1 MUX. This is somewhat challenging - so think what input information you need.

(b) Use contraction to solve the edge cases when i = 0, k 1. You do not have to simplify the internals of the MUX, just explain why you contracted as you did.