Time And Space Complexity In Machine Learning

If you are a Computer Science student or learning machine learning algorithms, then you must have come across these two most important terms. Let us understand Time Complexity and Space Complexity in more detail. Time Complexity and Space Complexity are two very important terms when developing an algorithm.

In this article, all the important details regarding time complexity and Space Complexity will be covered. Stay tuned to us till the end of the topic to get a complete understanding of these two most important terms in Computer Science.

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Time Complexity

It talks about the time required by an algorithm to execute completely.

Time complexity is a very important concept in machine learning, and you need to be very well aware of these most important concepts in computer science. It tells us how fast an algorithm can run and which approach is best out of the many possible ones.

In a simple way, the time complexity is the total amount of time required by the algorithm to complete its execution. We calculate time complexity by tracking fundamental operations like basic math’s, assignment operators, comparisons, etc.

It is helpful to know that the time complexity of an algorithm depends on how it is constructed and how much work it must perform. It is unconcerned with technical details like computer type, programming language, and other related issues.

This eliminates the need to consider various computer configurations when understanding and comparing algorithms. This idea is useful for comparing different algorithms and making smart choices when we want to make our computer programs run faster.

Space Complexity

It talks about the total space required by an algorithm.

Space complexity is knowing how much computer memory an algorithm or problem uses while it is running. This includes the memory for the data you are working with and the memory needed by the algorithm.

For example, let us picture a kitchen where we need room for storing many ingredients, such as utensils, food items, pots, bottles, and other important things that will take up space. Similarly, memory is required by the algorithm to store both data and the modules it works with.

Space complexity process to figure out how much memory an algorithm requires for smooth execution. We will try to use an approach or algorithm that uses less memory. Hence, knowing space complexity is important.

The memory needed while the algorithm is running is called the auxiliary space. It is similar to the space you require on your kitchen counter when you are cooking. The amount of memory an algorithm uses depends on how much input it is dealing with.

It is crucial to understand that space complexity and time complexity are two entirely different concepts. Space complexity is the amount of memory an algorithm uses, whereas time complexity is how quickly it operates. A fast algorithm does not necessarily mean that it uses less memory, and vice versa.

Tips To Reduce Time and Space Complexity

Some of the most important tips that must be kept in mind while writing an algorithm or code are given here. You can follow these steps to make a good algorithm to execute your code with less time and space complexity.

Understand the Problem Statement Properly

The first step before writing code is to understand the problem statement clearly. Try to read it more than once to completely understand the question being asked, and then give an efficient approach with effective time and space complexity.

Write an Algorithm First

Now, after understanding the problem statement, you need to develop an efficient algorithm that will solve your problem in the most effective way possible.

Develop the Algorithm Details

After writing the algorithm, try to implement it by adding the necessary steps required and you need to see if the problem statement has been solved. If the solution is obtained, you can work on the algorithm to optimize it more and make it more effective.

Reviewing the Optimized Code

At last, when the code has been fully optimized, the final step is to review it properly. The factors to be considered while reviewing are time and space complexity. Also, check if any more optimization is necessary in the final code.

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What Is The Importance of Time And Space Complexity?

Let us understand both time and space complexity with the help of a real-life example for better understanding.

Importance of Time Complexity

Suppose you are cooking a meal, and the time taken to finish cooking depends on the recipe’s complications and how fast you can work. In the same way, time complexity is all about how long a program takes to run on the computer. The less time it takes to run, the more efficient the code.

In programming, if we use an efficient approach, then our program will execute faster. It is essential to pick the right techniques and methods while writing code to make it more effective, which yields better performance.

Importance of Space Complexity

Let us suppose that we are shifting into a room and need to decide what we will put in the space available. If we do not properly plan it, then we will definitely use more space than we actually need, which is definitely not what we want.

Similarly, if we are not careful in computer programming, using unoptimized code can lead to the wastage of crucial memory space, making our code less effective and slow. But if we justify the use of our memory, then our code will work faster, and no wastage of space will occur. This is very important when we have very limited space, and we need our programs to work faster we need an optimized approach.

Now, we can clearly see how important both terms are in the field of Computer Science.

What Is The Use Of Asymptotic Notations?

Asymptotic notation is the mathematical notation that is used to tell how fast an algorithm will work as the size of the problem gets bigger. It helps us to check what the effect is on the size of the problem with respect to time.

We use asymptotic notations to compare two algorithms. It does not compare the two algorithms directly. Instead, it uses time complexity and space complexity as the base factors to compare two or more algorithms and the changes as we grow in size with respect to time.

Let us check the three main parts of the Asymptotic Notations used in Computer Science.

Big-Oh (O) notation

The big-Oh (O) notation was put forward by Paul Bachmann in the year 1894.

These notations represent the upper bound on the algorithm’s runtime. Hence, it shows the worst time of the algorithm and is used in the asymptotic analysis of an algorithm. In simple words, how bad it can get. It generally tells us the maximum amount of time required for the execution of an algorithm.

Big-Oh (O) Notations

According to 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

}

It will return the highest possible value for the input (big-O).

Big-Omega (Ω) notation

This notation is used to represent the floor rate of growth of a given function. Also, it is known for giving the best case for the given input. Unlike Big-O notation, it gives the lower bound of the run time of an algorithm. This condition enables the algorithm to execute in the shortest amount of time.

Big-Omega (Ω) Notations

According to the definition:

Ω (g(n)) =

{

f(n): if there exists a positive number c and N, such that f(n) >= cg(n) for all n >= N.

}

Big-Theta (Θ) notation

This notation lies between the below and above of the function. It helps in the analysis of the average-case time complexity of an algorithm. It stands for both the lower bound and the upper bound of run time of an algorithm. Hence, it yields us the average time complexity of an algorithm.

Big-Theta(Θ) Notations

According to the definition : f(n) is

Θ(g(n)) = {

if there exist positive numbers c1, c2 and N such that c1g(n) <= f(n) <= c2g(n),

for all n >= N

}

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

Let us have a look at the three most important cases in the asymptotic analysis of algorithms.

Best Case: As the name suggests, it takes minimum amount of time for the execution of code.
Worst Case: This shows the average time needed for the complete execution of the code.
Worst Case: This represents the maximum time required by the code for complete execution. We generally calculate the worst scenario to determine the effectiveness of a code.

How to Calculate Time Complexity

Let us understand how to calculate time complexity with a simple example code.

def add_numbers(x, y):

return x + y

a = 5

b = 10

result = add_numbers(a, b)

print(result)

Here, the function add numbers takes two parameters and returns the sum of two numbers as a result. Now let us check how to calculate its time complexity.

Let us look at the code and count the number of operations like addition and assignment operators.
Let us count number of times this action is performed. In this code, we have one addition operation and one assignment operation.
Now let us take the input size, which is of constant time, as O(1+1) = O(2).
This is nothing but a constant. Hence, we can consider it equal to the constant time, i.e., O(2) = O(1).
No matter how many times we add the numbers or perform arithmetic operations, it will have a constant time complexity.
In case of loops, the time complexity of algorithm increases.

Time Complexity of Searching Algorithms

Let us have a look on the time complexity of two most important searching algorithms.

Algorithm	Best case	Worst case
Linear search time complexity	O(1)	O(n)
Binary search time complexity	O(1)	O(log n)

Time Complexity Of Sorting Algorithms

Let us check the time complexity of some famous sorting algorithms.

Algorithm	Best case	Average case	Worst case
Bubble sort time complexity	Ω(n)	θ(n^2)	O(n^2)
Selection sort time complexity	Ω(n^2)	θ(n^2)	O(n^2)
Insertion sort time complexity	Ω(n)	θ(n^2)	O(n^2)
Merge sort time complexity	Ω(n log(n))	θ(n log(n))	O(n log(n))
Quick sort time complexity	Ω(n log(n))	θ(n log(n))	O(n^2)
Heap sort time complexity	Ω(n log(n))	θ(n log(n))	O(n log(n))

Space Complexity of Sorting Algorithms

Let us check the space complexity of searching algorithms from this table.

Algorithm	Space Complexity
Bubble Sort	1
Selection Sort	1
Insertion Sort	1
Merge Sort	n
Quick Sort	n
Heap Sort	1

Difference Between Time And Space Complexity

Let us check the important differences between the time and the space complexity through the table given below.

Time Complexity	Space Complexity
Calculates the required amount of time	Calculates the amount of memory space needed
All statements are measured in time.	All variables, inputs, and outputs are counted in terms of memory space.
The main factor is determined by the size of the input data.	Mainly determined by the auxiliary variable size
In terms of solution optimization, this is more significant	Much more significant for maximizing the solution

Time And Space Complexity FAQs

Q1. How are Time complexity and space complexity related to each other?

Ans: Time complexity and space complexity are inversely related to each other. If we try to improve one then we have to compromise the other.

Q2. What are some of the notations used in the asymptotic analysis of algorithms?

Ans: There are mainly three standard notations used in the asymptotic analysis of an algorithm.

Big-O notations (Worst-case complexity)
Big- Omega notations (Best-case complexity)
Theta Notations (Average time complexity)

Q3. Which is more important in asymptotic analysis? Time Complexity or Space complexity?

Ans: Both time and space complexity are equally important while calculating the asymptotic analysis.