Let ~s_i~ be the number of stamp types that friend ~i~ has and let ~t~ be the number of stamp types that Alice has. Observe that Alice satisfies the constraints if and only if ~s_i + t~ is odd for all ~i~. So the only important feature of a collection for this problem is the parity of the number of stamp types. Then Alice should either take the cheapest stamp type, or the cheapest two stamp types, or it is impossible. Checking which of these possibilities works can be easily done by checking the parity of each ~s_i~.

Time Complexity: ~\mathcal{O}(NK)~