Amortized Analysis: Banker's Method

An Innovative Approach to Dynamic Array Operations Is The Banker's Method

A crucial data structure in computer science is the dynamic array. They are a popular option for many applications because they make it possible to store and process enormous volumes of data effectively. Dynamic arrays can be expensive to maintain as they get larger, and adding elements to one can be expensive as well.

This issue is addressed with a revolutionary method for dynamic array operations called The Banker's Method.

The idea behind it is to charge a modest price for each inexpensive activity, like inserting an element, and then set aside those fees to pay for more expensive actions in the future. With this strategy, you save money each month for a major purchase, like a car, comparable to an amortizing loan.

The Banker's Method will be thoroughly examined in this essay, along with its underlying ideas, benefits, and practical applications.

A Dynamic Array Example Using the Banker's Method

Let's have a look at an instance of the Banker's Method being applied to a dynamic array.

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An empty array of size 0 and capacity 0 will be our starting point. In order to demonstrate how the Banker's Method actually functions, we shall add members to the array.

Including the array's initial element (a):
An empty array of size 0 and capacity 0 is where we begin. In order to use PushBack(a), we must first allocate a size 1 array, point to it, and then insert a value into the array. We also attach a token to so that, if necessary, we can pay to move it later.

Including the array's second element (b):
We must allocate a larger array because there won't be room for b in the current array when we use PushBack(b). We update the array after paying for relocating a with the token that was earlier placed on a. Then, for a cost of 1, we add b to the array. We placed one token on element b and one token earlier in the array because we still have two more tokens to spend (a).

Including the array's third element (c):
We must allocate a new array and copy over a and b when we call PushBack(c). We update the array after paying for relocating a and b with the tokens that were previously placed on them. Then, for a cost of 1, we add c to the array.

An Illustration of the Banker's Approach
Let's have a look at an illustration to see how the Banker's Method functions in real life. Starting with an empty dynamic array, we will add the entries "a," "b," and "c."

We must allocate a new array of size 1 and insert element "a" when we insert the first element. Then, we attach a token to "a" as payment for subsequently shifting it to a different array.

There isn't room in the existing array for the second member, "b," when we insert it, so we must create a larger array and transfer "a" to it. We pay for transferring 'a' by using the token we placed on it before, and after that, we insert 'b' into the new array. To pay for upcoming procedures, we placed a token on "b" and another token on "a."

Finally, we must allocate a new array, copy over elements "a" and "b," and then put element "c" into the new array. With the tokens we previously placed on 'a' and 'b,' we pay for shifting them to the new array.

Benefits of the Banker's Approach

The Banker's Method has a number of benefits over conventional dynamic array operations, including:

1. Lower cost of insertions: We can keep the cost of each insertion low even as the array gets bigger by collecting a tiny fee for each insertion and storing those charges for later.
2. Resource utilization that is more effective: By paying for the transfer of components to a new array with tokens that have been saved, we can reduce the number of array resizes necessary and utilize memory and other resources more effectively.
3. Improved performance: The Banker's Method can enhance the overall performance of dynamic array operations, making them quicker and more effective. This is done by keeping the cost of each insertion low.