Before getting into Big O Notation fastest to slowest time complexity, it’s always better to have an idea about what is Asymptotic analysis and its fundamentals. It will be really beneficial when analysing asymptotic notation in data structure with example and it will sharpen your big o coding knowledge for big o notation practice problems.

### Significance of Asymptotic Notation in data structures and algorithms

Asymptotic analysis based on calculating the running time of an operation on a mathematical basis. Basically in an algorithm, the mathematical relation of run time performance can be identified by asymptotic analysis.

#### Why do we use asymptotic notation?

By using the asymptotic notation in algorithm analysis, the time required by an algorithm to execute can be calculated and it’s fall under three categories.

### Asymptotic notation and its types

- Best case ( Omega Notation ) – Minimum time required for a programme to execute.
- Average case ( Theta Notation ) – Average time required for a programme to execute.
- Worst case ( Big O notation ) – Maximum time required for a programme to execute.

#### What are the factors that affect the running time of an algorithm?

- Speed of the computer
- The efficiency of the programming language
- Compiler performance
- Number of lines and loops in the code
- Number of operations within a loop

### What is Big Omega notation?

Big omega notation expresses the algorithm takes at least a certain amount of time to implement. Big omega is the asymptotic lower bound of an algorithm.

For **Ω** (**f(n)**), the running time will be at least **k . f(n)** for a constant **k**.

### What is Big O notation?

Big O notation explained the maximum time that the algorithm will take to implement. Analyzing an algorithm with big o notation is useful for predicting most level that the running time can be extended to. Big O notation is the asymptotic upper bound of an algorithm. It’s essential to find Big O of a function before getting an idea about its time complexity.

For **O (f(n))**, the running time will be at most **k . f(n)** for a constant k.

Modern computer science-based applications are highly using big o notation.Computer science industry applications mainly using the Big O notation to analyse algorithms and how they affect computational complexity.Big o Notation is using in order to calculate time and space requirements with respect to the increase of the input size of the algorithm.

### What is Big Theta notation?

Big theta notation expresses the average exact running time performance which lies in between the upper bound and the lower bound of an algorithm. Big theta notation is the asymptotically tight bound of an algorithm.

For **Θ(f(n))**, the running time lies in between the least value **k _{1} . f(n**) and the most value

**k**) for some constants

_{2}. f(n**k**and

_{1}**k**.

_{2}### Big O Notation fastest to slowest time complexity

**The formal definition of Big O:** Big O algorithm mainly gives an idea of how complex an operation is. It expresses how long time an operation will run concerning the increase of the data set which clearly describes the asymptotic time complexity.

**1 < log(n) < √n < n < n log(n) < n² < n³ < 2 ^{n }< 3^{n } < n^{n} **

A clarification of Big O Notation fastest to slowest ( best to worst complexity ) according to increasing order of asymptotic complexity can be implemented.

### Big O Complexity Chart

Big O function | Increasing behavior of time complexity |

O(1) | constant |

O(log n) | logarithmically change |

O(n) | linear change |

O(n log n) | Linearithmic change |

O(n²) | Quadratic change |

O(n³) | Cubic change |

O(2)^{n} | exponential change |

O(n!) | Factorial change |

- O(
**1**) – speed is constant. It doesn’t change with the size of the data set. - O(
**log n**) – Big O**log n**operation changes logarithmically. Little more time consuming than O(1) - O(
**n**) – linear change. Time taken is directly proportional to the size of the data set. - O(
**n log n**) – Linearithmic change in**n log n**big o operation. Time increases by a multiplier of 1 with the size of the data set doubles. - O(
**n²**) – Big O quadratic change can be seen. Time consumes four times more when the size of the data set doubles. - O(
**n³**) – Cubic change. Time consumes eight times more when the size of the data set doubles - O(
**2**) – exponential change. time takes twice long for every new element added to the data set.^{n} - O(
**n!**) – Factorial change. n! Times longer time consuming directly proportional to the size of the data set. - O(
**n**) – Highest time-consuming function and the slowest to implement because of its time complexity.^{n}

This is valid for larger data sets. However **n!** can place anywhere between **1** < **n!** < **n ^{n} ** with the change of the value of

**n**.

Big O notation order according to its time complexity can be analyzed clearly by the big o notation diagram below

**What is the importance of Big O notation in programming?**

Working with large data sets is very difficult and there are many things to concern about before selecting a data structure and an algorithm to use. The Big O notation expresses the time complexity which gives an idea about the time taken to perform a certain algorithm and also the space complexity which expresses the amount of space that will be required by an algorithm.

**Big O vs Little O: the difference between big O notation and little o notation in asymptotic notations**

Little-o is a stronger condition than a big-O. Big O can be explained as the inclusive upper bound, while little o can be explained as the strict upper bound.

**What is Little omega Notation?**

**Little omega Notation** is mainly written as ω, is a notation to describe the lower bound (which is not asymptotically tight) of the growth rate on the runtime of an algorithm.

### 9 Basic things you must know before a big o coding interview

- Should be able to define the importance of asymptotic notation in algorithm analysis
- Explain various asymptotic notations with example: Best case, average case and the worst case of an algorithm
- Factors that affect the running time of an algorithm
- Define big omega notation
- Define Big O notation understanding big o notation
- Define Theta notation
- Big O Notation fastest to slowest time complexity chart
- Difference between big O notation and little o notation in asymptotic notations
- What is Little omega Notation?

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