Problem 5. Ordering functions (20 points) Sort the following 25 functions from asymptotically smallest to asymptotically largest, indicating ties if there are any. You do not need to turn in proofs (in fact, please don't turn in proofs 421gn lg (n/8)+g lg lgn cos(n) +2 gn 1 15n 154 nl/lg lg n lg lg n 201 nlg n To simplify notation put all the functions in a single column, where f(n) being above g(n) in the column means f(n) = o(g(n)) If f(n) = (g(n), then write f(n) and g(n) separated by a comma on the same line. For example, the functions n2, n, (2), n3 should be written . n2 (a) Hint: When considering two functions f(n) and g(n) it is sometimes useful to consider the functions log f(n) and logg(n), since log f(n) = o(log g(n)) implies f(n) = o(g(n)) (but be careful, the implication does not work in the other direction, and in particular log f(n) = O(log g(n)) does NOT imply f(n) = O(g(n)))