 
Summary: COMPLEXITY: Exercise No. 1
due in two weeks
1. Prove or disprove:
(a) (2n)! = O(n! 2 ).
(b) f(n) = O(n) ) 2 f(n) = O(2 n ).
(c) log(n!) = (n log n).
(d) If f = o(g) then there is h such that f = o(h) and h = o(g).
(e) If f 1 ; f 2 ; : : : ; f k 2 O(n) then
P k
i=1
f i (n) = O(n).
(f) If f 1 ; f 2 ; : : : 2 O(n) then
P n
i=1
f i (n) = O(n 2 ).
(g) There is a function f such that f(n) = O(n 1+ ) for every > 0 but f(n) = !(n).
(h) If f is monotone positive function,f(n) = O(n) and f(n) 6= o(n) then f(n) = (n).
2. Let M be a ktape DTM with input and output which always stops with time complexity
t(n) and space complexity s(n). Prove that there is a constant c (which depends on M) such
that t(n) nc s(n) (assume s(n) > 0 for all n).
