For your solution of the mandatory exercise it might be convenient to assume that the weights are sorted as w_1 >= w_2 >= … >= w_n. This does not lose generality since the (1+1) EA does not care (more precisely, it has the same stochastic behavior) when you rename the bits.

Note that the exercise considers a class of functions, which includes, amongst others, OneMax and BinVal. So you cannot assume that all weights are different. As a matter of fact, you are not allowed to change the algorithm depending on the function.

Finally, make sure to prove that your partition is fitness-based (in particular verify condition 4 from the definition) and prove that for every search point from a level there is a way to improve to a higher level. Most likely you need an argument like: If the search point is in level L_i (which might be defined according to the fitness only), then I still know that the search point has this-and-that kind of structure, and therefore there is a mutation that results in a search point from a higher level.

Best regards,

Carsten