c) Let’s develop a runtime recursion for an algorithm that computesfi(3by3blocks
7. (1 level)Optional bonus problem: More medians(This is anoptionalchallenge problem. It is not the most effective way to raise your grade in the course.Only solve it if you want an extra challenge.)We saw in class a randomized algorithm for computing the median, where the expected running time wasO(n). Design a algorithm for computing the median where theworstcaserunning time isO(n).Hint: It’s all in finding a good pivot.
Pseudocode
:
CS 170, Fall 2016, HW 2
8
1:
function
S
ELECT
(
k
,
A
)
2:
n
←
size of
A
3:
if
n
≤
100
then
4:
L
←
sorted version of
A
5:
return
L
[
n
/
2
]
6:
M
←
empty list of size
n
/
5
7:
for
i
=
0
,
1
,...,
n
/
5

1
do
8:
M
[
i
]
←
median of
A
[
5
i
,
5
i
+
4
]
9:
m
←
Select
(
n
/
10
,
M
)
10:
A
low
←
elements of
A
less than
m
11:
A
high
←
elements of
A
greater than
m
12:
r
←
rank of
m
in
A
(compute in linear time)
13:
if
k
≤
r
then
14:
return
Select
(
k
,
A
low
)
15:
else
16:
return
Select
(
k

r
,
A
high
)
Running time analysis
:
The algorithm does linear work to recur on a list of size
n
/
5 and 3
n
/
4. This yields a recursion of
T
(
n
)
≤
T
(
3
n
/
4
)+
T
(
n
/
5
)+
n
Writing out the recursion tree yields a bound of
T
(
n
)
≤
n
+(
3
/
4
+
1
/
5
)
n
+(
3
/
4
+
1
/
5
)
2
n
+
...
=
O
(
n
)
as desired.