1. Textbook p. 58: 3-2 all
a.
and
For
=
=
=
= 0
b.
nk = O(cn) and o(cn)
For
=
=
=
= 0
c. None of them applies.
For nsin n oscillates between n (> n1/2) and 1/n (< n1/2).
Hence no C or n0 can be found for O, ,
o, or .
d.
2n/2 = O(2n) and
o(2n).
For
= 0
since
2n = 2n/2 2n/2.
e.
,
,
and
, where m should read c.
For they are equal by one of the properties of
function.
f.
,
,
and
.
For
from p.55 and
.
Proving
is quite complex involving integrals.
See for example p. 9 of S. Baase, Computer Algorithms, 2nd ed., Addison-Wesley, 1988.
2. Textbook p.75:
4.3 - 1 all
a. Since a = 4 and b = 2,
.
Hence this is case 1 of Master Theorem.
Hence
.
b. Since a = 4 and b = 2,
.
Hence this is case 2 of Master Theorem.
Hence
.
c. Since a = 4 and b = 2,
.
Also c = 1/2 satisfies the inequality condition of case 3 of Master Theorem
because
.
Hence this is case 3 of Master Theorem.
Hence
.
4.3 - 3
Since a = 1 and b = 2,
.
Hence
.
Hence this falls into case 2 of Master Theorem.
Hence
.
4.3 - 5
Take
.
Check to see that this satisfies all the conditions of case 3 of Master Theorem
except the regularity condition.