Asymptotic Analysis | What Is Asymptotic Analysis?

As we think about the analysis of algorithms, asymptotic analysis enters our minds. We will discuss asymptotic analysis here since it is fundamental to algorithms. Previously, we had learned this information from the previous guide. Now let’s make a clear assumption regarding the asymptotic study.


What is Asymptotic Analysis?

During asymptotic analysis of algorithms, the mathematical basis of algorithmic performance is defined. The technique of estimating an algorithm’s running time in terms of mathematical units to explore its limitations, or “run-time performance,” is known as asymptotic analysis.

Based on this study, we can determine the best and worst-case scenarios and average processing times for the task. Although an asymptotic method in the analysis is not a method of deep learning training, it is an important diagnostic tool for assessing an algorithm’s efficiency rather than its correctness. As an example of asymptotic analysis:

When the value of n gets large, the asymptotic behavior of f(n) (such as f(n)=c*n or f(n)=c*n2) is the growth of f(n). We usually ignore low values of n since the program’s performance is mainly affected by input sizes. The better the algorithm, the slower its asymptotic growth rate.


Asymptotic Notations 

Asymptotic notations describe how long an algorithm runs as the input tends towards a certain value. As an example: Bubble sort takes the best case time when the input array has been sorted, or linearly if the array is already sorted. On the other hand, when the array is in reverse order, sorting takes the longest time or the worst-case scenario. In most cases, it takes average time whether the input array is sorted or in reverse order. A representation of these durations is based on asymptotic notations.

Asymptotic notation can be expressed in three ways.


  1. Big-O Notation


Algorithms are expressed using Big-O notation to express their maximum running times. Therefore, it reflects the complexity of an algorithm under the worst-case situation.


O(g(n)) = { f(n): there exist positive constants c and n0

            such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 }


This expression is an example of f(n) belonging to the set O(g(n)) if, for sufficiently large (n), there is a positive constant (C) which lies between (0) and CG(n).

When (n) is any value, the running time of an algorithm doesn’t exceed that of O(g(n)).


  1. Omega Notation


Omega notation indicates the lower bounds of an algorithm’s running time. This determines the complexity of the algorithm in the best case.


Ω(g(n)) = { f(n): there exist positive constants c and n0 

            such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 }


The above expression can be viewed as a function f(n) which belongs to the set *(g(n)) if there exists a positive constant (c) which lies above CG(n), for sufficiently big (n).


The minimum time required by the algorithm for any value of (n) is Omega Ω(g(n)).


  1. Theta Notation


Using theta notation, you can include both the above and below functions. Average-case complexity analysis is conducted using it since it is used for calculating the upper and lower bounds of the algorithm’s run-time.


For a function g(n), θ(g(n)) is given by the relation:


θ(g(n)) = { f(n): there exist positive constants c1, c2 and n0

            such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0 }


In general, a function f(n) belongs to the set θ(g(n)), if (c1) and (c2) are positive constants that can easily fall between c1g(n) and c2g(n), for sufficiently large n.


If the asymptotic tight bound of a function f(n) lies between c1g(n) and c2g(n) for all n ≥ n0, then f(n) is described as asymptotically tight.


Asymptotic Efficiency


The asymptotic efficiency of an unbiased estimator increases as the sample size expands. The term “asymptotically efficient estimator” refers to an estimator that has an asymptotic efficiency of 1.0. It is generally believed that an asymptotically efficient estimator will tend to the theoretical limit as the sample size increases.


How to Analyze Algorithm?

Following are the steps involved in analyzing the running time of an algorithm:

  1. Put the algorithm into practice completely since algorithm analysis practice is extremely important.

2. Estimate how long each basic operation will take.

3. Calculate the unknown quantities that describe how frequently the basic operations are executed.

4. Prepare a realistic model of the input into the program.

5. In case of unknown quantities, analyze them assuming the modeled input.

6. To determine the running time, multiply the time by the frequency for each operation.


Using classic algorithm analysis on early computers, it was possible to predict running times exact to decimal seconds.

Final Note


By observing and measuring how much memory and time an algorithm consumes, an asymptotic analysis can determine its efficiency. Following that, asymptotic notations can be used to analyze algorithm behavior in terms of run-time. These allow us to express run-time as mathematical equations, allowing us to do our tasks with the greatest efficiency and minimal effort.




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