In computer science, big O notation is a way to measure the efficiency of an algorithm. The "O" in big O notation stands for "order of magnitude." In other words, it's a way to compare the efficiency of an algorithm to the efficiency of another algorithm.

There are four different types of big O notation:

• O(1) – This is the most efficient possible algorithm. It doesn't matter how big the input is, the algorithm will always take the same amount of time to run.

• O(log n) – This is an algorithm that is slightly less efficient than an O(1) algorithm. It will take more time to run as the input gets bigger, but the amount of time it takes to run will increase at a slower rate than an O(1) algorithm.

• O(n) – This is an algorithm that is less efficient than an O(log n) algorithm. It will take more time to run as the input gets bigger, and the amount of time it takes to run will increase at a faster rate than an O(log n) algorithm.

• O(n2) – This is an algorithm that is less efficient than an O(n) algorithm. It will take more time to run as the input gets bigger, and the amount of time it takes to run will increase at a much faster rate than an O(n) algorithm.

To calculate the big O notation of an algorithm, you need to look at how the **algorithm behaves as the input size** gets larger. For example, let's say you have an algorithm that sorts an array of numbers. If the array has 10 numbers in it, the algorithm might take 1 second to sort the array. But if the array has 100 numbers in it, the algorithm might take 10 seconds to sort the array. And if the array has 1,000 numbers in it, the algorithm might take 100 seconds to sort the array.

As you can see, the time it takes to sort the array increases as the array gets bigger. But it doesn't increase linearly. It increases at a much slower rate. In fact, it increases at a logarithmic rate.

So, the big O notation of this algorithm is O(log n).

There are two other important things to know about big O notation:

• The big O notation only applies to the worst case scenario

## What is big O notation?

In computer science, big O notation is used to describe the computational complexity of an algorithm in terms of the amount of time it takes to execute as a function of the size of the input. In other words, it describes how an algorithm scales as the size of the input data increases.

There are two main types of big O notation: worst case and average case. Worst case big O notation gives an **upper bound on the execution time** of an algorithm, while average case big O notation gives a more realistic estimate of the execution time.

Big O notation is a way of formalizing the intuition that some algorithms are more efficient than others. It is not a perfect measure of efficiency, but it is a useful tool for comparing the relative efficiency of different algorithms.

There are four main types of complexity that can be described using big O notation: constant, logarithmic, linear, and polynomial.

Constant complexity is the simplest kind of complexity. An algorithm with constant complexity will execute in the same amount of time regardless of the size of the input.

Logarithmic complexity is slightly more complex. An algorithm with logarithmic complexity will take longer to execute as the size of the input increases, but the increase will be less dramatic than with an algorithm of linear complexity.

Linear complexity is the most common type of complexity. An algorithm with linear complexity will take longer to execute as the size of the input increases, but the increase will be directly proportional to the size of the input.

Polynomial complexity is the most complex type of complexity. An algorithm with polynomial complexity will take longer to execute as the size of the input increases, but the increase will be more dramatic than with an algorithm of linear complexity.

The big O notation is a way of describing the worst case scenario for an algorithm. It is not a perfect measure of efficiency, but it is a useful tool for comparing the relative efficiency of different algorithms.

## What is an example of big O notation?

There are many examples of big O notation, but one of the most common is the time complexity of an algorithm. Time complexity is a measure of how long an algorithm takes to run, and is typically expressed as a function of the input size. For example, if an algorithm takes 30 seconds to run on an input of size n, its time complexity is O(n).

## How do you calculate big O notation?

In computer science, big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends to a particular value or infinity. It is usually expressed as a simple fractional power of the variable.

The big O notation is often used to express the **time complexity or space complexity** of an algorithm. Time complexity is a **well-studied field in computer science**, and is usually expressed in terms of the number of operations that an algorithm performs in relation to the size of the input. Space complexity is a related field, and is usually expressed in terms of the number of memory cells that an algorithm uses in relation to the size of the input.

Big O notation is a way of formalizing the intuitive idea that some problems are "harder" than others. For example, a problem that can be solved in polynomial time is considered "easier" than a problem that can only be solved in exponential time. The big O notation gives a precise definition of what it means for a problem to be "harder" than another.

The big O notation is often used in conjunction with the little o notation and the Ω notation. These three notions together are called the Landau symbols.

Big O notation is a mathematical notation that is used to describe the limiting behavior of a function. The function is usually a power function of the variable. The big O notation is often used to descri**be the time complexity or space com**plexity of an algorithm.

## What is the time complexity of an algorithm?

When discussing the time complexity of an algorithm, we are referring to the amount of time it takes for the algorithm to run to completion. There are two types of time complexity: worst-case and best-case. Worst-case time complexity is the amount of time it takes for the algorithm to complete when the input is at its worst possible state. Best-case time complexity is the amount of time it takes for the algorithm to complete when the input is at its best possible state.

There are four different time complexity classes: constant, logarithmic, linear, and polynomial. Constant time complexity means that the algorithm will always take the same amount of time to complete, no matter the input. Logarithmic time complexity means that the algorithm will take a logarithmic amount of time to complete, based on the size of the input. Linear time complexity means that the algorithm will take a linear amount of time to complete, based on the size of the input. Polynomial time complexity means that the algorithm will take a polynomial amount of time to complete, based on the size of the input.

The time complexity of an algorithm can be determined by analyzing the pseudo-code of the algorithm. The time complexity can also be determined by running the algorithm on different inputs and measuring the amount of time it takes to complete.

## What is the space complexity of an algorithm?

The space complexity of an algorithm is the amount of memory required to run the algorithm. This is typically measured in terms of the number of bytes required.

Space complexity is important to consider when choosing an algorithm, as it can have a significant impact on the amount of memory required to run a program. For example, a **sorting algorithm with a space complexity** of O(n) requires only a few bytes of memory to sort a list of n items, while an algorithm with a space complexity of O(n2) requires n2 bytes of memory.

There are two types of space complexity: static and dynamic. Static space complexity is the amount of memory required by the algorithm when it is run on a given input. Dynamic space complexity is the amount of memory required by the algorithm during its execution.

The space complexity of an algorithm can be affected by the input size, the data structures used, and the order in which the algorithm processes the input. For example, an algorithm that sorts a list of n items using a linked list will have a lower space complexity than an algorithm that sorts the same list using an array, because the linked list requires less memory to store the data.

Time complexity is often confused with space complexity, but they are two distinct measures. Time complexity is the amount of time required to run an algorithm, while space complexity is the amount of memory required.

algorithm, the time complexity is the number of operations required to execute the algorithm. Space complexity is the number of memory locations required to store the data used by the algorithm.

## What is the worst-case time complexity of an algorithm?

There is no definitive answer to this question as it depends on a number of factors, including the specific algorithm in question, the resources available to the algorithm, and the specific inputs used. However, in general, the worst-case time complexity of an algorithm is the amount of time required for the algorithm to complete its task when given the worst possible input. This worst-case time complexity can be expressed as a function of the input size ( typically denoted by n). For example, if an algorithm has a worst-case time complexity of O(n2), this means that the algorithm will take at most n2 time units to complete its task when given the worst possible input. The time complexity of an algorithm is often determined by analyzing the algorithm's running time on different inputs of different sizes. However, worst-case time complexity can also be determined by analyzing the algorithm's worst-case running time on any input.

## What is the best-case time complexity of an algorithm?

What is the best-case time complexity of an algorithm?

This is a **difficult question to answer due** to the fact that there are a multitude of factors that can affect the answer. However, we can narrow it down to a few key points.

The best-case time complexity of an algorithm is the number of operations that the algorithm can perform in the best-case scenario. This number is usually determined by the input size. For example, if an algorithm has a best-case time complexity of O(n), this means that the algorithm can perform at most n operations in the best-case scenario.

The best-case time complexity of an algorithm is often determined by the worst-case time complexity of the algorithm. This is because the worst-case time complexity is a more reliable measure of an algorithm's performance. However, there are some algorithms whose best-case time complexity is not determined by the worst-case time complexity. For example, algorithms that use heuristics can have a best-case time complexity that is different from the worst-case time complexity.

There are a few factors that can affect the best-case time complexity of an algorithm. The first is the input size. If an algorithm has a best-case time complexity of O(n), this means that the algorithm can perform at most n operations in the best-case scenario. The second is the algorithm itself. Some algorithms are more complex than others and this can affect the best-case time complexity. The third factor is the environment in which the algorithm is run. This can include the hardware, the operating system, and the user.

The best-case time complexity of an algorithm is an important measure of an algorithm's performance. It is often used to compare algorithms and to choose the best algorithm for a given problem.

## What is the average-case time complexity of an algorithm?

There is no definitive answer to this question as it depends on a variety of factors, including the specific algorithm in question and the inputs it is provided with. However, in general, the average-case time complexity of an algorithm is the amount of time it takes the algorithm to run on average given a set of inputs. This can be contrasted with the worst-case time complexity, which is the amount of time the algorithm takes to run on the worst possible input.

There are a number of ways to measure the time complexity of an algorithm. One common approach is to count the number of **elementary operations the algorithm performs**. This approach works well for simple algorithms, but can be difficult to use for more complex algorithms. Another approach is to measure the running time of the algorithm on a given set of inputs. This approach is more accurate, but can be difficult to use for complex algorithms.

In general, the average-case time complexity of an algorithm is the amount of time it takes the algorithm to run on average given a set of inputs. This can be contrasted with the worst-case time complexity, which is the amount of time the algorithm takes to run on the worst possible input. There are a number of ways to measure the time complexity of an algorithm. One common approach is to count the num**ber of elementary operations the algorithm p**erforms. This approach works well for simple algorithms, but can be difficult to use for more complex algorithms. Another approach is to measure the running time of the algorithm on a given set of inputs. This approach is more accurate, but can be difficult to use for complex algorithms.

## What is the worst-case space complexity of an algorithm?

There is no definitive answer to this question as it depends on the specifics of the algorithm in question. However, some generalisations can be made.

The worst-case space complexity of an algorithm is the maximum amount of space that the algorithm needs to complete its task, under any circumstances. This is typically determined by the size and complexity of the inputs to the algorithm.

For example, consider a sorting algorithm that takes in a list of numbers and sorts them in ascending order. The worst-case space complexity of this algorithm would be the maximum amount of space that is required to store the list of numbers, regardless of how many numbers are in the list.

Similarly, the **worst-case space complexity of a graph traversal algorithm** would be the maximum amount of space that is required to store the graph, regardless of its size or complexity.

In general, the worst-case space complexity of an algorithm is determined by its inputs. The larger and more complex the inputs, the more space the algorithm will need to complete its task.

## Frequently Asked Questions

### What is Big O notation and why does it matter?

Big O notation compares the time it takes an algorithm to run against the number of inputs. If an algorithm takes a long time to run or uses a lot of memory, then it has a high Big O rating.

### What is the best way to understand Big O in code?

The best ways to understand Big O in code are through examples, as well as understanding its meaning and implications.

### What is an example of Big O?

The Big O notation is most often used for functions that take a long time to run or perform a lot of work. For example, an algorithm that increments each number in a list of length n might be said to have an "O(n) time" runtime, meaning it takes roughly the same amount of time to run as it does to list the numbers one by one.

### What are the different types of Big O algorithms?

There are three main types of Big O algorithms: the logarithmic, the linear, and the polynomial.

### What does Big O notation tell you about an algorithm?

Big O notation tells you how many operations an algorithm performs.

## Sources

- https://lankydan.dev/2017/04/23/learning-big-o-notation-with-on-complexity/
- https://www.bigocheatsheet.com/
- https://www.freecodecamp.org/news/big-o-notation-why-it-matters-and-why-it-doesnt-1674cfa8a23c/
- https://jarednielsen.com/big-o-omega-theta/
- https://www.wikihow.com/Calculate-the-Day-of-the-Week
- https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-o-notation
- https://www.programiz.com/dsa/asymptotic-notations
- https://en.wikipedia.org/wiki/Big_O_notation
- https://mathworld.wolfram.com/Little-ONotation.html
- https://www.coursera.org/lecture/algorithmic-toolbox/big-o-notation-j5bev
- http://web.mit.edu/16.070/www/lecture/big_o.pdf
- https://pages.cs.wisc.edu/~vernon/cs367/notes/3.COMPLEXITY.html

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