# AMORTIZED the amortized cost. Ø Different amortized costs being

AMORTIZED ANALYSIS OF A
BINARY COUNTER

There are
two types of Binary Counter:

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1.     Aggregate Method

2.     Accounting Method

3.     Potential Method

Accounting Method:

Ø Accounting method in a amortized analysis
is a method which is based on accounting.

Ø It is the easiest way to calculate
the amortized cost of an operation.

Ø It is better than the aggregate
method or potential method.

Ø But this method do not give us a
guarantee of obvious analysis.

Ø This Method does not include any
complexity.

Ø Each operation in this method is
assigned with a amortized cost

Ø The actual cost should be greater
than the amortized cost.

Ø Different amortized costs being
assigned to multiple different amortized operations. Some working contain
amortized cost more or less than the actual cost.

Ø When the cost exceeds from the actual
cost of amortized operation. Then the specific objects is assigned with a
difference in a data structure called credit.

Ø If the amortized cost is less than
the actual cost, then the credit is used to pay for those operations.

Ø Overall amortized cost of operations
should be >= overall actual cost total credit should be >= 0.

Ø The credit in the amortized analysis
is linked with a data structure.

·
The
accounting method itself includes two operations:

1.     Stack Operation

2.     Binary Operation

STACK OPERATION: The stack operation consists of three
operations:

1.
Push

2.
Pop

3.
Multi-pops

§  To learn the counting method of
amortized analysis , here is the example:

Actual Costs:

Push
1,

Pop       1,

Multi pop   min (a, p)       where a
is argument given to the multi pop and p is size of a stack

Now, the assigned amortized cost

Push
2,

Pop      0,

Multi pop 0,

Now we can see that , the amortized cost of multi pop is 0 whereas the actual
cost was a variable , so the overall amortized cost of three operation is O(1). But the overall cost of amortized
operation is O(n).

But sometimes the amortized cost can be change asymptotically.

Binary Counter Increment:

The binary counter with the increment operation starts with zero. The
running time of increment operation is directly proportional to the no.of bits
flipped.

Example:
Let us take an example of a Dollar bill to present each unit of cost.

For amortized analysis , let
charge an amortized cost of 2 dollars to set it to a 1 dollar. If the bit is
set , we will take 1 dollar out 2 dollars to pay actual bit , then we place the
left bit as a credit. At the anyother time we can use that left 1 dollar , and
our bit will never reset to zero. We will just pay for the dollar reset bill.
Thus the number of 1’s in the counter can never be negative neither the amount
of credit can be negative. So for “n” increment operators the amount of total
amortized cost is O(n), in which actual cost is bound.

AMORTIZED ANALYSIS OF A
BINARY COUNTER

There are
two types of Binary Counter:

4.     Aggregate Method

5.     Accounting Method

6.     Potential Method

Accounting Method:

Ø Accounting method in a amortized analysis
is a method which is based on accounting.

Ø It is the easiest way to calculate
the amortized cost of an operation.

Ø It is better than the aggregate
method or potential method.

Ø But this method do not give us a
guarantee of obvious analysis.

Ø This Method does not include any
complexity.

Ø Each operation in this method is
assigned with a amortized cost

Ø The actual cost should be greater
than the amortized cost.

Ø Different amortized costs being
assigned to multiple different amortized operations. Some working contain
amortized cost more or less than the actual cost.

Ø When the cost exceeds from the actual
cost of amortized operation. Then the specific objects is assigned with a
difference in a data structure called credit.

Ø If the amortized cost is less than
the actual cost, then the credit is used to pay for those operations.

Ø Overall amortized cost of operations
should be >= overall actual cost total credit should be >= 0.

Ø The credit in the amortized analysis
is linked with a data structure.

·
The
accounting method itself includes two operations:

3.     Stack Operation

4.     Binary Operation

STACK OPERATION: The stack operation consists of three
operations:

4.
Push

5.
Pop

6.
Multi-pops

§  To learn the counting method of
amortized analysis , here is the example:

Actual Costs:

Push
1,

Pop       1,

Multi pop   min (a, p)       where a
is argument given to the multi pop and p is size of a stack

Now, the assigned amortized cost

Push
2,

Pop      0,

Multi pop 0,

Now we can see that , the amortized cost of multi pop is 0 whereas the actual
cost was a variable , so the overall amortized cost of three operation is O(1). But the overall cost of amortized
operation is O(n).

But sometimes the amortized cost can be change asymptotically.

Binary Counter Increment:

The binary counter with the increment operation starts with zero. The
running time of increment operation is directly proportional to the no.of bits
flipped.

Example:
Let us take an example of a Dollar bill to present each unit of cost.

For amortized analysis , let
charge an amortized cost of 2 dollars to set it to a 1 dollar. If the bit is
set , we will take 1 dollar out 2 dollars to pay actual bit , then we place the
left bit as a credit. At the anyother time we can use that left 1 dollar , and
our bit will never reset to zero. We will just pay for the dollar reset bill.
Thus the number of 1’s in the counter can never be negative neither the amount
of credit can be negative. So for “n” increment operators the amount of total
amortized cost is O(n), in which actual cost is bound.