Tutorial Playlist

Data Structure Tutorial

Overview

Arrays in Data Structures: A Guide With Examples

Lesson - 1

All You Need to Know About Two-Dimensional Arrays

Lesson - 2

All You Need to Know About a Linked List in a Data Structure

Lesson - 3

The Complete Guide to Implement a Singly Linked List

Lesson - 4

The Ultimate Guide to Implement a Doubly Linked List

Lesson - 5

The Fundamentals for Understanding Circular Linked List

Lesson - 6

The Ultimate Guide To Understand The Differences Between Stack And Queue

Lesson - 7

Implementing Stacks in Data Structures

Lesson - 8

Your One-Stop Solution for Stack Implementation Using Array

Lesson - 9

Your One-Stop Solution for Queue Implementation Using Array

Lesson - 10

Your One-Stop Solution to Learn Depth-First Search(DFS) Algorithm From Scratch

Lesson - 11

Your One-Stop Solution for Stack Implementation Using Linked-List

Lesson - 12

The Definitive Guide to Understand Stack vs Heap Memory Allocation

Lesson - 13

All You Need to Know About Linear Search Algorithm

Lesson - 14

All You Need to Know About Breadth-First Search Algorithm

Lesson - 15

A One-Stop Solution for Using Binary Search Trees in Data Structure

Lesson - 16

The Best Tutorial to Understand Trees in Data Structure

Lesson - 17

A Complete Guide to Implement Binary Tree in Data Structure

Lesson - 18

A Holistic Look at Using AVL Trees in Data Structures

Lesson - 19

All You Need to Know About Tree Traversal in Data Structure

Lesson - 20

The Best Guide You’ll Ever Need to Understand B-Tree in Data Structure

Lesson - 21

The Best Guide You'll Ever Need to Understand Spanning Tree in Data Structure

Lesson - 22

The Best and Easiest Way to Understand an Algorithm

Lesson - 23

Your One-Stop Solution to Understand Shell Sort Algorithm

Lesson - 24

Your One-Stop Solution to Quick Sort Algorithm

Lesson - 25

The Most Useful Guide to Learn Selection Sort Algorithm

Lesson - 26

Everything You Need to Know About Radix Sort Algorithm

Lesson - 27

Everything You Need to Know About the Counting Sort Algorithm

Lesson - 28

Everything You Need to Know About the Merge Sort Algorithm

Lesson - 29

Insertion Sort Algorithm: One-Stop Solution That Will Help You Understand Insertion Sort

Lesson - 30

Everything You Need to Know About the Bubble Sort Algorithm

Lesson - 31

The Best Guide You’ll Ever Need to Understand Bucket Sort Algorithm

Lesson - 32

Your One-Stop Solution to Understand Recursive Algorithm in Programming

Lesson - 33

The Definitive Guide to Understanding Greedy Algorithm

Lesson - 34

Your One-Stop Solution to Understand Backtracking Algorithm

Lesson - 35

The Fundamentals of the Bellman-Ford Algorithm

Lesson - 36

Your One-Stop Solution for Graphs in Data Structures

Lesson - 37

The Best Guide to Understand and Implement Solutions for Tower of Hanoi Puzzle

Lesson - 38

A Simplified and Complete Guide to Learn Space and Time Complexity

Lesson - 39

All You Need to Know About the Knapsack Problem : Your Complete Guide

Lesson - 40

The Fibonacci Series: Mathematical and Programming Interpretation

Lesson - 41

The Holistic Look at Longest Common Subsequence Problem

Lesson - 42

The Best Article to Understand What Is Dynamic Programming

Lesson - 43

A Guide to Implement Longest Increasing Subsequence Using Dynamic Programming

Lesson - 44

A Holistic Guide to Learn Stop Solution Using Dynamic Programming

Lesson - 45

One Stop Solution to All the Dynamic Programming Problems

Lesson - 46

Understanding the Fundamentals of Binomial Distribution

Lesson - 47
A Simplified and Complete Guide to Learn Space and Time Complexity

In this tutorial, you will explore computational complexity (space and time complexity), developed by Juris Hartmanis and Richard E. Stearns to assess the difficulty of an algorithm. As you all know, human nature strives to find the most efficient way to complete their daily tasks. The overarching thought process behind innovation and technology is to make people's lives easier by providing solutions to problems they may face. In the world of computer science and digital products, the same thing occurs. To perform better, you need to write algorithms that are time efficient and use less memory.

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What Is Time Complexity?

Time complexity is a type of computational complexity that describes the time required to execute an algorithm. The time complexity of an algorithm is the amount of time it takes for each statement to complete. As a result, it is highly dependent on the size of the processed data. It also aids in defining an algorithm's effectiveness and evaluating its performance.

What Is Space Complexity?

When an algorithm is run on a computer, it necessitates a certain amount of memory space. The amount of memory used by a program to execute it is represented by its space complexity. Because a program requires memory to store input data and temporal values while running, the space complexity is auxiliary and input space.

What Does It Take To Develop a Good Algorithm?

what-is-space-and-time-complexity

A good algorithm executes quickly and saves space in the process. You should find a happy medium of space and time (space and time complexity), but you can do with the average. Now, take a look at a simple algorithm for calculating the "mul" of two numbers.

Step 1: Start.

Step 2: Create two variables (a & b).

Step 3: Store integer values in ‘a’ and ‘b.’ -> Input

Step 4: Create a variable named 'mul'

Step 5: Store the mul of 'a' and 'b' in a variable named 'mul" -> Output

Step 6: End.

You will now see how significant space and time complexity is after understanding what they are.

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How Significant Are Space and Time Complexity?

Significant in Terms of Time Complexity

The input size has a strong relationship with time complexity. As the size of the input increases, so does the runtime, or the amount of time it takes the algorithm to run.

Here is an example.

Assume you have a set of numbers S= (10, 50, 20, 15, 30)

There are numerous algorithms for sorting the given numbers. However, not all of them are effective. To determine which is the most effective, you must perform computational analysis on each algorithm.

Significant-of-time-and-space-complexity

Here are some of the most critical findings from the graph:

  • This test revealed the following sorting algorithms: Quicksort, Insertion sort, Bubble sort, and Heapsort.
  • Python is the programming language used to complete the task, and the input size ranges from 50 to 500 characters.
  • The results were as follows: "Heap Sort algorithms performed well despite the length of the lists; on the other hand, you discovered that Insertion sort and Bubble sort algorithms performed far worse, significantly increasing computing time." See the graph above for the results.
  • Before you can run an analysis on any algorithm, you must first determine its stability. Understanding your data is the most important aspect of conducting a successful analysis.

What Are Asymptotic Notations?

Asymptotic Notations are programming languages that allow you to analyze an algorithm's running time by identifying its behavior as its input size grows. This is also referred to as an algorithm's growth rate. When the input size increases, does the algorithm become incredibly slow? Is it able to maintain its fast run time as the input size grows? You can answer these questions thanks to Asymptotic Notation.

You can't compare two algorithms head to head. It is heavily influenced by the tools and hardware you use for comparisons, such as the operating system, CPU model, processor generation, and so on. Even if you calculate time and space complexity for two algorithms running on the same system, the subtle changes in the system environment may affect their time and space complexity.

As a result, you compare space and time complexity using asymptotic analysis. It compares two algorithms based on changes in their performance as the input size is increased or decreased.

Asymptotic notations are classified into three types:

  1. Big-Oh (O) notation
  2. Big Omega ( Ω ) notation
  3. Big Theta ( Θ ) notation

Now, go over each of these notations one by one.

1. Big-Oh (O) Notation

Paul Bachmann invented the big-O notation in 1894. He inadvertently introduced this notation in his discussion of function approximation.

From the definition: O (g(n)) = 

{

f(n) : there exist positive constant c and n0 such that 0 <= f(n) <= c*g(n) 

For all n >= n0

}

'n' denotes the upper bound value. If a function is O(n), it is also O(n2) and O(n3).

It is the most widely used notation for Asymptotic analysis. It specifies the upper bound of a function, i.e., the maximum time required by an algorithm or the worst-case time complexity. In other words, it returns the highest possible output value (big-O) for a given input.

Big-oh-notation-in-space-and-time-complexity

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2. Big-Omega (Ω) notation

Big-Omega is an Asymptotic Notation for the best case or a floor growth rate for a given function. It gives you an asymptotic lower bound on the growth rate of an algorithm's runtime.

From the definition: The function f( n ) is Ω (g(n)) if there exists a positive number c and N, such that f(n) >= cg(n) for all n >= N.

omega-notation-in-space-and-time-complexity.

3. Big-Theta (Θ) notation

Big theta defines a function's lower and upper bounds, i.e., it exists as both, most, and least boundaries for a given input value.

From the definition : f(n) is Θ(g(n)) if there exists positive numbers c1, c2 and N such that c1g(n) <= f(n) <= c2g(n) for all n >= N.

theta-notation-in-space-and-time-complexity.

Best Case, Worst Case, and Average Case in Asymptotic Analysis

Best Case: It is defined as the condition that allows an algorithm to complete statement execution in the shortest amount of time. In this case, the execution time serves as a lower bound on the algorithm's time complexity.

Average Case: You add the running times for each possible input combination and take the average in the average case. Here, the execution time serves as both a lower and upper bound on the algorithm's time complexity.

Worst Case: It is defined as the condition that allows an algorithm to complete statement execution in the shortest amount of time possible. In this case, the execution time serves as an upper bound on the algorithm's time complexity.

You will now see how to calculate space and time complexity after grasping the significance of space and time complexity.

Significant in Terms of Space Complexity

Space complexity refers to the total amount of memory space used by an algorithm/program, including the space of input values for execution. Calculate the space occupied by variables in an algorithm/program to determine space complexity.

However, people frequently confuse Space-complexity with auxiliary space. Auxiliary space is simply extra or temporary space, and it is not the same as space complexity. To put it another way,

Auxiliary space + space use by input values = Space Complexity

The best algorithm/program should have a low level of space complexity. The less space required, the faster it executes.

Method for Calculating Space and Time Complexity

Methods for Calculating Time Complexity

To calculate time complexity, you must consider each line of code in the program. Consider the multiplication function as an example. Now, calculate the time complexity of the multiply function:

  1. mul <- 1
  2. i <- 1
  3. While i <= n do
  4.      mul = mul * 1
  5.      i = i + 1
  6. End while

Let T(n) be a function of the algorithm's time complexity. Lines 1 and 2 have a time complexity of O. (1). Line 3 represents a loop. As a result, you must repeat lines 4 and 5 (n -1) times. As a result, the time complexity of lines 4 and 5 is O. (n).

Finally, adding the time complexity of all the lines yields the overall time complexity of the multiple function fT(n) = O(n).

The iterative method gets its name because it calculates an iterative algorithm's time complexity by parsing it line by line and adding the complexity.

Aside from the iterative method, several other concepts are used in various cases. The recursive process, for example, is an excellent way to calculate time complexity for recurrent solutions that use recursive trees or substitutions. The master's theorem is another popular method for calculating time complexity.

Methods for Calculating Space Complexity

With an example, you will go over how to calculate space complexity in this section. Here is an example of computing the multiplication of array elements:

  1. int mul, i
  2. While i < = n do
  3.    mul <- mul * array[i]
  4.    i <- i + 1
  5. end while
  6. return mul

Let S(n) denote the algorithm's space complexity. In most systems, an integer occupies 4 bytes of memory. As a result, the number of allocated bytes would be the space complexity.

Line 1 allocates memory space for two integers, resulting in S(n) = 4 bytes multiplied by 2 = 8 bytes. Line 2 represents a loop. Lines 3 and 4 assign a value to an already existing variable. As a result, there is no need to set aside any space. The return statement in line 6 will allocate one more memory case. As a result, S(n)= 4 times 2 + 4 = 12 bytes.

Because the array is used in the algorithm to allocate n cases of integers, the final space complexity will be fS(n) = n + 12 = O (n).

As you progress through this tutorial, you will see some differences between space and time complexity.

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Time Complexity vs. Space Complexity

You now understand space and time complexity fundamentals and how to calculate it for an algorithm or program. In this section, you will summarise all previous discussions and list the key differences in a table.

                  Time Complexity

                    Space Complexity

Calculates the time required

Estimates the space memory required

Time is counted for all statements

Memory space is counted for all variables, inputs, and outputs.

The size of the input data is the primary determinant.

Primarily determined by the auxiliary variable size

More crucial in terms of solution optimization

More essential in terms of solution optimization

Now that you have reached the end of the tutorial on space and time complexity, sum up what you’ve learned thus far.

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Next Steps

In this tutorial, you learned what exactly space and time complexity are and how significant they are. You then learned how to calculate space and time complexity, and, finally, you learned the difference between space and time complexity.

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Do you have any queries about this tutorial on space and time complexity? Please leave them in the comments section at the bottom of this page if you do. Our experts will be happy to respond to your questions as earliest as possible!

About the Author

Soni UpadhyaySoni Upadhyay

Soni Upadhyay is with Simplilearn's Research Analysis Team. She's a Computer Science and Engineering graduate. Programming languages are her area of expertise. She has a brilliant knowledge of C, C++, and Java Programming languages.

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