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Selected Solutions for Chapter 17:
Amortized Analysis
Solution to Exercise 17.13
Let
c
i
D
cost of
i
th operation.
c
i
D
(
i
if
i
is an exact power of 2
;
1
otherwise
:
Operation
Cost
1
1
2
2
3
1
4
4
5
1
6
1
7
1
8
8
9
1
10
1
:
:
:
:
:
:
n
operations cost
n
X
i
D
1
c
i
±
n
C
lg
n
X
j
D
0
2
j
D
n
C
.2n
N
1/ < 3n :
(Note: Ignoring floor in upper bound of
P
2
j
.)
Average cost of operation
D
Total cost
# operations
< 3
.
By aggregate analysis, the amortized cost per operation
D
O.1/
.
Solution to Exercise 17.22
Let
c
i
D
cost of
i
th operation.
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Selected Solutions for Chapter 17: Amortized Analysis
c
i
D
(
i
if
i
is an exact power of 2
;
1
otherwise
:
Charge each operation $3 (amortized cost
y
c
i
).
±
If
i
is not an exact power of 2, pay $1, and store $2 as credit.
±
If
i
is an exact power of 2, pay $
i
, using stored credit.
Operation
Cost
Actual cost
Credit remaining
1
3
1
2
2
3
2
3
3
3
1
5
4
3
4
4
5
3
1
6
6
3
1
8
7
3
1
10
8
3
8
5
9
3
1
7
10
3
1
9
:
:
:
:
:
:
:
:
:
:
:
:
Since the amortized cost is $3 per operation,
n
X
i
D
1
y
c
i
D
3n
.
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 Spring '10
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