AMORTIZED ANALYSIS OF A

BINARY COUNTER

There are

two types of Binary Counter:

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.