Big O Notation Explained: Time and Space Complexity

10 Feb 2023


All of us refer to the steps involved in cooking our favourite recipe. We explore multiple ways to cook it but finally follow those steps that confirm the fastest cooking method with the least effort. The process is analogous to studying the algorithms designed and finally choosing the best based on the quantifiable measures regarding time. This article gives an insight about what is Big O notation in Data structure. Also, after referring this article, the readers will be able to define Big O notation.

Interesting facts about Big O Notation

The Big O Notation is the most significant member of the notation family discovered by Paul Bachmann and Edmund Landau and identified as Bachmann–Landau notation or asymptotic notation.

What is Big O Notation in Data structure?

In data structures, Big Order Notation refers to as Big O Notation and is the most effective expression to compare the functioning and performance of the algorithms. Programmers and Developers use this as a Master theorem to evaluate the efficiency of an algorithm in terms of the amount of time required to run the algorithm, the resources consumed by the algorithm, and the impact on its execution based on its input parameters.

Big O Notation, often expressed as O(n) eliminates the constant parameters to facilitate the approximation of the time consumed by an algorithm in reference to X, where X tends to infinity. This asymptotic expansion is a numerical metric to express the performance of an algorithm with respect to the time and space complexities. Though the notation potentially denotes the amount of memory used by the algorithm, it is not capable of assessing the speed of the algorithm. The objective of the notation is to provide a clear understanding of the best, average, and maximum amount of time or the worst-case time complexity of an algorithm.

In other words, the Big O notation counts the number of operations processed by an algorithm. Thus, the notation indicates the maximum number (the upper time limit) required to implement the algorithm. This notation is a formal way of expressing the asymptotic pattern of the algorithm and is termed Asymptotic Notation. The mathematical annotation describes the relationship between the input and the conduct of the algorithm. As the volume of the input data increases, the time required to execute the algorithm increases.

Before diving into the DSA, a programmer must understand what is Big O notation in Data structure. Also should be able to calculate Big O notation in various cases. 

How is Big O determined?

Algorithm analysts use the combination of time and space complexities to determine the Big O for an algorithm. They follow the following steps to determine the time complexity-

  • Split each function in the algorithm into a single instruction

  • Precisely estimate the worst-case time complexity for each of the instructions.

  • Calculate the total of all these time complexities by ignoring the constant parameters.

  • The highest term is the Big Order of the algorithm.

The space complexity is analyzed based on the following steps –

  • Consider the language used to code the program based on the algorithm

  • Study the size of the inputs and the memory locations required to store them

  • Calculate the total of all these memory locations and treat the number as Space complexity

Big O Complexity Analysis

Based on the function designed in each instruction of an algorithm, there are six Time complexities and two attributes of Space complexities associated with the Big O.

What is Time Complexity

The time complexity can be defined as the amount of time required by an algorithm to execute each statement of code until it is finished. It is denoted as a function that depends on the size of the input. The big O notation is the most popular way to express how time-consuming an algorithm is. The time complexity examples describes the need to calculate the time complexity for algorithms. The following table is an illustration of time complexity examples.

Constant Time Complexity: O(1)

Constant Time Complexity is the simplified complexity of all complexities. Regardless of the input, if the algorithm utilizes the same time frame to execute the instructions, the complexity is constant and expressed as O(1). Further, irrespective of the size of the inputs in the algorithm, if the algorithm consumes the same amount of memory space, the space complexity is O(1). An instruction to compare two elements is an example of this complexity.

Logarithmic Time Complexity: O(logn)

Logarithmic Time Complexity is observed where the number of inputs expands logarithmically affecting the running time of an algorithm and expressed as O(log n). The logarithmic time complexity is always less than the polynomial time complexity Algorithms for binary search generally demonstrate this complexity.

Linear Time Complexity: O(n)

Linear Time Complexity is a function expressing a direct proportion between the time required to execute the algorithm and the number of inputs. As the number of inputs increases, the space to hold the inputs increases, and the time to process multiple inputs increases. In such cases, the complexity is expressed as o(n) where n is the number of inputs. Finding the location of an element in a linear array is an example of this complexity.

Linearithmetic Time complexity: O(nlogn)

Linearithmetic Time complexity a combination of linear and arithmetic is an intermediate complexity that focuses on its linear time complexity. The complexity expressed as O(n log n) increases arithmetically as the number of inputs achieves the highest number. 

Polynomial Time Complexity: O(n^c)

Polynomial Time Complexity is determined when an algorithm is in the form of a linked list or may process multiple values as a single data element. This complexity is subdivided into Quadratic and Cubic Time complexities.

When two loops are nested in such a way that the time of execution of the loops proportionately influences the square of the number of inputs, n, Quadratic time complexity is evaluated and expressed as O(n^2) or O(n*n). Typically, Insertion sorting algorithms show this complexity.

Similarly, when three loops are nested in such a way that the time of execution of the loops proportionately influences the cube of the number of inputs, n, Cubic time complexity is calculated and expressed as O(n^3) or O(n*n*n). Usually, algorithms handling 3D parameters or three-dimensional matrices exhibit this complexity.

Exponential Time Complexity: O(c^n)

Exponential Time Complexity annotated as O(c^n) is seen if the complexity of an algorithm directly impacts cn where n is the number of inputs and c is a constant. As a conventional practice, c=2 in most applications. An algorithm to generate a password is one of the best examples of this complexity.

Factorial Time Complexity: O(n!)

Factorial Time Complexity expressed as O(n!) explores all the possible loops or conditions in the algorithm. The value of the Factorial Time complexity is always more than the Exponential time complexity. Algorithms with recursions illustrate this complexity.

What is Space Complexity?

The term space complexity of the algorithm refers to how much memory the algorithm needs to solve a particular problem. The amount of space an algorithm uses to run as a function of input length is measured by the space complexity.

Unit Space Complexity

Unit Space Complexityexpressed as O(n) where n is one, indicates the most efficient algorithm with minimum memory and execution time.

Multiple Space Complexity

Multiple Space Complexity indicates the auxiliary or the storage space and the temporary space required by the algorithm during its execution. The complexity is expressed as O(n) where n is the sum of the number of bytes or size of the temporary and auxiliary storage space. As the intermediary storage elements in an algorithm increase, its complexity increases.

Why Big O is important?

  • The Big O Notation defines the transformability or scalability of the algorithm. It helps developers to understand the algorithm in-depth and explore the reusability of the algorithm for voluminous data.

  • The Big O notation provides the analysis irrespective of the loops or the constructs used in the algorithm.

  • The Big O Notation gives a clear idea of the conduct of the algorithm in algebraic terms. The notation prioritizes the dominant aspects of the algebraic expression and thus encourages decision-making for the programmers.

Best, Average, and Worst Complexity

The complexity of an algorithm is studied based on its performance and classified as the best case, average case, and worst case. An algorithm is the best when it executes the least number of steps on the given ‘n’ number of input data. On the other hand, if the algorithm runs the maximum number of instructions on the given ‘n‘ number of input data, it is considered the worst case. Normally, when an algorithm executes the average number of steps on the given ‘n’ number of input data, it is considered an average case. A programmer can rarely estimate the average case of an algorithm due to the non-availability of the required information.


This article explains what is Big O notation in Data structure with examples. The Big O Notation is a fundamental tool for programmers and developers and acts as a guide to them as it directs them to the best algorithm easing them with the coding. The mathematical formulation traces the rate of change of the proficiency of an algorithm when its inputs expand. Time complexity can be further understood thru time complexity examples.

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