CS170 (Fall 2014) –
Solutions
for
Homework 3
Krishna Parashar, SID 23664968,
cs170-hm
, Collaborators: None.
September 19, 2014
1. Practice with recurrence relations
(a)
F
(
n
) =
θ
(
n
)
(b)
G
(
n
) =
θ
(log
n
)
(c)
H
(
n
) =
θ
(log
n
)
(d)
I
(
n
) =
θ
(
n
)
(e)
J
(
n
) =
θ
(
n
0
.
5
)
(f)
K
(
n
) =
θ
(
n
)
(g)
L
(
n
) =
θ
(
n
2
.
4
)
(h)
M
(
n
) =
θ
(2
n
)
1

CS170–Fall 2014 Homework 3 Krishna Parashar,
cs170-hm
2
2. Procedural Terrain Generation
(a) Everytime we call our Algorithm we branch into 4 factors, and we make each of these
problems of size
n/
2
All the cool kids know that we can use a Recurrence Relations to get this super awesome
relation :
T
(
n
) = 4
T
n
2
+
O
(1)
And then, oh gentle reader, we apply the Master Theorem to get these values:
a
= 4
, b
= 2
, d
= 0
We use these values for our logrithm:
log
2
4 = 2
Since, according to the Master Theorem, it is trivial to prove that our value of 2 is
greater than 0 (it is after all how numbers work), and thus we end up with the solution:
θ
(
n
2
)
(b) Unfortunetly, even if we could reduce the other operations (such as the computation