CS 473
HeadBanging Session 0 (January 20–21, 2009)
Spring 2009
CS 473: Undergraduate Algorithms, Spring 2009
Head Banging Session 0
January 20 and 21, 2009
1.
Solve the following recurrences. If base cases are provided, find an
exact
closedform solution.
Otherwise, find a solution of the form
Θ(
f
(
n
))
for some function
f
.
•
Warmup:
You should be able to solve these almost as fast as you can write down the answers.
(a)
A
(
n
) =
A
(
n

1
) +
1, where
A
(
0
) =
0.
(b)
B
(
n
) =
B
(
n

5
) +
2, where
B
(
0
) =
17.
(c)
C
(
n
) =
C
(
n

1
) +
n
2
(d)
D
(
n
) =
3
D
(
n
/
2
) +
n
2
(e)
E
(
n
) =
4
E
(
n
/
2
) +
n
2
(f)
F
(
n
) =
5
F
(
n
/
2
) +
n
2
•
Real practice:
(a)
A
(
n
) =
A
(
n
/
3
) +
3
A
(
n
/
5
) +
A
(
n
/
15
) +
n
(b)
B
(
n
) =
min
0
<
k
<
n
(
B
(
k
) +
B
(
n

k
) +
n
)
(c)
C
(
n
) =
max
n
/
4
<
k
<
3
n
/
4
(
C
(
k
) +
C
(
n

k
) +
n
)
(d)
D
(
n
) =
max
0
<
k
<
n
D
(
k
) +
D
(
n

k
) +
k
(
n

k
)
, where
D
(
1
) =
0
(e)
E
(
n
) =
2
E
(
n

1
) +
E
(
n

2
)
, where
E
(
0
) =
1 and
E
(
1
) =
2
(f)
F
(
n
) =
1
F
(
n

1
)
F
(
n

2
)
, where
F
(
0
) =
1 and
F
(
2
) =
2
?
(g)
G
(
n
) =
n G
(
p
n
) +
n
2
2.
The
Fibonacci numbers
F
n
are defined recursively as follows:
F
0
=
0,
F
1
=
1, and
F
n
=
F
n

1
+
F
n

2
for every integer
n
≥
2. The first few Fibonacci numbers are 0,1,1,2,3,5,8,13,21,34,55,
....
Prove that any nonnegative integer can be written as the sum of distinct
nonconsecutive
Fibonacci numbers. That is, if any Fibonacci number
F
n
appears in the sum, then its neighbors
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 Fall '08
 Chekuri,C
 Algorithms, Fibonacci number

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