The.Algorithm.Design.Manual.Springer-Verlag.1998

Exercises
(b) If I prove that an algorithm takes worst-case time, is it possible that it takes O(n) on all
inputs?
(c) If I prove that an algorithm takes worst-case time, is it possible that it takes O(n) on
some inputs?
(d) If I prove that an algorithm takes worst-case time, is it possible that it takes O(n) on all
inputs?
(e) Is the function , where for even n and for odd
n?
4. For each of the following, answer yes, no, or can't tell. Explain your reasoning!
(a) Is ?
(b) Is ?
(c) Is ?
(d) Is ?
5. (*) Give a proof or counterexample to the following claim: if f(n) = O(F(n)) and g(n) = O(G(n)),
then f(n)/g(n) = O(F(n)/G(n)).
6. (*) Does f(n) = O(g(n)) imply that ? Explain your reasoning!
7. (*) Give a proof or counterexample to the following claim: for all functions f(n) and g(n), either
f(n) = O(g(n)) or g(n) = O(f(n)).
8. (*) When you first learned to multiply numbers, you were told that means add x a total of y
times, so . What is the time complexity of multiplying two n-digit
numbers in base b (people work in base 10, of course, while computers work in base 2) using the
repeated addition method, as a function of n and b. Assume that single-digit by single-digit
addition or multiplication takes O(1) time. (hint: how big can y be as a function of n and b?)
9. (*) In grade school, you learned to multiply long numbers on a digit by digit basis, so that
. Analyze the time complexity of multiplying
two n-digit numbers with this method as a function of n (assume constant base size). Assume that
single-digit by single-digit addition or multiplication takes O(1) time.
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