# Amortized Analysis: Physicist's Method

## Recognizing the Physicist's Amortized Analysis Method

A method for figuring out an algorithm's average-case time complexity is amortized analysis. It is a crucial tool in computer science and by giving programmers a clear idea of the cost of operations over time, it can aid in the creation of algorithms that are more effective. This article's main method for undertaking amortized analysis is the physicist's method.

According to the physicist's method, a data structure's state can be translated into an integer that represents its potential using a potential function.

This idea is comparable to what you learnt about potential energy in high school physics. For instance, a ball's potential energy rises as it is carried to the top of a hill; when the ball is let to roll down the hill, however, its prospective energy falls while its kinetic energy rises. The data structure of the physicist's technique is used in the same way, with the potential function serving as a placeholder for potential future work.

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#### The Potential Function's Definition

Two requirements must be met by the potential function: first, the potential of the data structure's initial state (phi of h sub 0) must equal 0, and second, the potential can never be negative (phi of h sub t is greater than or equal to 0).

The amortized cost of an operation can be determined when the possible function has been identified. The genuine cost (c sub t) plus the difference in potential before and after the operation (phi(h sub t) - phi(h sub t-1)) equal the amortized cost.

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#### Selecting the Possible Function

The correct potential function must be selected in order to employ the physicist's method for amortized analysis properly. The potential should rise if the actual cost of an operation is low in order to accumulate funds for subsequent activities. The potential should decrease to cover the cost of the work being done, however, if the genuine cost is high.

#### True Cost to Amortized Cost Ratio

The definition of amortized cost is (c sub I + phi(h sub I - phi(h sub i-1)), and it can be used to calculate the relationship between the total actual costs of all operations and the total amortized costs. The potential function's positive and negative values will cancel out, leaving only the beginning potential (phi of h sub 0) and the end potential. This is vital to keep in mind (phi of h sub n).

The physicist's approach is an effective instrument for amortized analysis and figuring out an algorithm's average-case time complexity. The amortized cost of operations can be determined and linked to the true cost by creating a possible function and selecting it wisely. Understanding this approach is crucial for creating effective algorithms and enhancing computer systems' performance.

Updated: Aug 04, 2023