Time Complexity: Significance, Types, Algorithms

Time Complexity: Significance, Types, Algorithms

Published
25th Apr, 2024
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15 Mins

We all have heard the quote, “Time is Money!”. In this article, we are going to discuss time complexity of algorithms and how they are significant to us. As a software developer, I have been building applications and I know how important it becomes for us to deliver solutions that are fast and efficient. Nobody would want to use a system which takes a lot of time to process large input size. We would want to make use of data structures in organizing and storing data in a way so that it can be accessed and manipulated efficiently. It defines the way data is arranged in a computer's memory and the operations that can be performed on that data. Time complexity for data structures is important aspect while developing software solutions and is implemented in most of the programming languages.

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What is Time Complexity?

Time complexity means the number of times the statement is executed. The Time complexity of an algorithm is the actual time needed to execute the particular codes. Also, it depends on various factors like a Programming language, operating system, processing power, and so on. The idea behind time complexity is that it can only measure the algorithm's execution time in a way that is dependent solely on the algorithm and its input.

The "Big O notation" evaluates an algorithm's time complexity. The Big O notation is a language used to describe an algorithm's time complexity. It is how we compare the efficacy of various approaches to a problem and helps us make decisions.

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How to Calculate Time Complexity?

Let's take an example of time complexity to understand the problem and how it works with the algorithm.

E.g., Imagine a room of 100 kids in which you gave your pencil to one person. You have to find that pencil without knowing to whom you gave it. To demonstrate those problems, let's understand.

Here are some methods for locating the pencil and determining the O order.

• O(n2): You go to the first child in the class and ask him if he has the pencil. You also ask these kids about the other 99 students in the classroom, such as whether they have that pencil and so on.
• O(n): Going around and asking each student individually is acceptable (N)
• O(log n): Now I divide the class into two groups and ask, "Is it on the left or right side of the classroom?" After that, I divide that group into half and ask again, and so on. Repeat the process until you only have one student left with your pencil. This is what we called as  O(log n).

Let us see some other examples and processes to find the time complexity algorithm.

E.g. Let us consider a model machine that uses the following specifications:

• Single processor
• 32-bit
• Linear execution
• 1 unit time for arithmetic and logical operation
• 1 unit time for assignment and return operation

Example: Find the sum of the two numbers.

With pseudo code it will be : sum(a, b) {return a + b }

In the above-given example, the code will take 2 units of time. One for arithmetic and the other for return. Thus, total cost of performing operation is (Tsum) = 1 + 1 = 2.

Therefore, Time Complexity  = O(2)  = O(1).

Significance of Time Complexity?

Let's first understand what an algorithm is.

In computer programming, an algorithm is a finite sequence of well-defined instructions that are typically executed in a computer to solve a class of problems or perform a common task. According to the definition, there must be a sequence of defined instructions given to the computer in order for it to execute an algorithm/perform a specific task. Variations in the way the instructions are defined may occur in this context. A specific set of instructions can be defined in any number of ways to perform the same task.

The algorithm would be executed in a computer, which leads to the next variation in terms of the operating system, processor, hardware, and so on that are used, all of which can impact how an algorithm is executed. Given that various factors can influence an algorithm's outcome, it is prudent to comprehend how effectively such programs are used to perform a task. To determine this, we must assess an algorithm's Space Complexity and Time complexity.

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An algorithm's space complexity determines how much space or memory it requires to run as a function of the length of the input. While the time complexity of an algorithm calculates how much time requires to complete a particular function on a particular input length. In other words, the time complexity is the amount required to execute each source code statement program in an algorithm. When a statement is set to run repeatedly, the number of times it is run is N multiplied by the time it takes to run that function each time.

Types of Time Complexity

Big O Notation expresses an algorithm's run time in terms of how quickly it grows in comparison to the input 'n' by defining the N number of operations performed on it. Thus, an algorithm's time complexity is denoted by the sum of all O[n] assigned to each line of function.

There are various types of time complexities used; let's go over them one by one:

1. Constant Time Complexity: O(1)

When an algorithm has constant time with order O (1) and is independent of the input size n, it is said to have constant time with order O (1).

Let's see some cases here to understand the constant time complexity

Find if a number is odd or even

```function EvenOrOdd(n) {
return n % 2 ? 'Odd' : 'Even';
}
console.log(EvenOrOdd(10)); // => Even
console.log(EvenOrOdd(10001)); // => Odd ```

Note:- you could also replace the n % 2 with the bit AND operator: n & 1. if the first bit is 1then it is odd or else even.

2. Linear Time Complexity: O(n)

Linear Complexity O(n) means the algorithm takes longer to complete the particular task as the input grows.

Let's say we want to find the maximum value from an unsorted array.

```function findMaximum(n) {
let max;
let counter = 0;
for (let i = 0; i < n.length; i++) {
counter++;
if(max === undefined || max < n[i]) {
max = n[i];
}
}
console.log(`n: \${n.length}, counter: \${counter}`);
return max;
} ```

It will check every element from n. If the current item is more significant, max will try to do the assignment.

3. Logarithmic Time Complexity: O(log n)

When an algorithm shrinks the input data size in each step, it is said to have a logarithmic time complexity. This indicates that the number of operations is not equal to the size of the input. As the size of the input increases, the number of operations decreases. Binary trees and binary search functions contain algorithms. The search for a given value in an array is accomplished by splitting the array in two and beginning the search in one split. This ensures that the operation is not performed on every data element.

A non-linear time complexity algorithm is one whose running time increases non-linearly (n2) with the length of the input. Nested loops, in general, follow this order, where one loop takes O(n), and if the function involves a loop within a loop, it follows the O(n)*O(n) = O(n2) order.

Similarly, if the function defines 'm' loops, the order is given by O (n m), which are known as polynomial time complexity functions.

5. Exponential Time Complexity: O(2^n)

The growth rate in exponential time algorithms doubles with each addition to the input (n), iterating through all subsets of the input elements. Every time an input unit increases by one, the number of operations performed doubles.

Time Complexity of Sorting Algorithms

Exploring the time complexities of sorting algorithms aids us in selecting the best sorting technique in a given situation. Let's understand what is meant by "Sorting".

It is an algorithm that arranges the elements of a list in a certain order[ either ascending or descending]. The output is a permutation of the reordering of the input.

Here are some sorting methods:

1. Selection Sort

In the selection sort, the smallest element is exchanged with the first element of the unsorted list of elements. Then the second smallest element is exchanged with the second element of the unsorted list of elements, and so on, until all the elements are sorted.

The time complexity of Selection Sort in the best case is O(n^2). In the worst case, the time complexity is O(1).

Let's understand how the selection sort works.

Consider here an example of an array: arr[] = {23,56,12,78};

First Move:

1. In the first position in the sorted array, the whole array is traversed from starting index 0 to 4 linearly. In the first position where 23 is stored, after traversing the whole array, it comes to mind that 12 is the lowest value.
2. Therefore, it replaces 23 with 12, the lowest value in the array. After, it presents the first index in the sorted list.

Second Move:

1. For the second position, 56 are present. Again, it will be traversed with the other array list in a sequential manner.
2. After traversing the element, we found that 23, the lowest index value in the array, should be placed in the second position, so swap the values.

Third Move:

1. Now in the third place, we can see that 56 is the lowest compared to 78. So it will be placed in the third position.
2. So, the resulting array is the sorted array.

2. Merge Sort

Merge is an example of the divide and conquers strategy method. It firstly divides the array into equal half and then combines them in a sorted way. It is a recursive algorithm that continuously splits an array equally. Merge sort is the process of taking two sorted arrays and combining them together into a single, sorted, new array. The best case is O(nlogn). In the worst case, the time complexity of the merge sort is O(nlogn).

3. Bubble Sort

Bubble sort is the simplest sorting algorithm. Bubble sort is the algorithm that works by repeatedly swapping the values between two numbers. Suppose they are present in the wrong order. Bubble sorting in computer graphics is popular because it can detect small errors.

4. Quick Sort

Quick sort is the famous algorithm among comparison-based sorting algorithms. Like merge sort, quick sort uses the divide-and-conquer technique, a recursive algorithm. Quick sort uses the divide and conquers technique to gain the advantages of the merge sort.

5. Heap Sort

It is a comparison-based sorting algorithm and is part of the selection sort family. Although somewhat slower in practice on most machines than a good implementation of Quick sort, it has the advantage of a more favorable worst-case O(nlogn) runtime.

6. Bucket Sort

Bucket sort also imposes restrictions on the input to improve performance. It works well if the input is drawn from a fixed set. It is the generation of Counting sort.

7. Insertion Sort

Insertion sorts algorithm elements one by one and places them in the right position where it belongs in the sorted list of elements. Insertion sort is a simple and efficient comparison sort. In this algorithm, each iteration removes an element from the input list and inserts it into the sorted sub-list. The choice of the element being removed from the input list is random, and this process is repeated until all input elements have gone through.

Time Complexity Notation: Types

Here are some of the asymptotic Notations for calculating the time complexity are as follows:

1. Big-O (O) Notation:-  It is used to express an algorithm's maximum allowable running time. As a result, it provides worst-case time complexity.

For example, consider the insertion sort. It takes linear time in the best and quadratic time worst cases. So, we can say that the time complexity for insertion sort is O(n^2).

2. Big Omega (Ω) Notation:- The systematic way to express the lower bound of an algorithm's running time is to use the notation Ω. It calculates the base time complexity.

Let us consider the same insertion sort again. The time complexity of an insertion sort is written as Ω(n). It is not useful for insertion sort, as we are looking for the worst and average cases.

3. Big Theta (Θ) Notation:-  The systematic way to express the lower limit and upper bound of an algorithm's running time is to use the notation (n).

4. Little Oh (O) Notation:- Except for the Big-Oh, Big-Omega, and Big-Theta notations, there are some other notations. One of them is the little o notation. An upper bound that cannot be tight is described using little o notation. In other words, f has a broad upper bound (n).

Let f(n) and g(n) be the mapping functions for positive real numbers. The function f(n) is said to be o(g(n)) if, for any real positive constant c, there exists an integer constant n0 1 such that f(n) > 0.

5. Little Omega (Ω) Notation:- Little Omega (Ω) represents a rough estimate of the growth order, whereas Big Omega (Ω) may represent the exact order of growth. Notation is used to represent a lower bound that is not asymptotically tight.

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Time Complexity with Examples

Time complexity, or computational complexity refers to the amount of time taken by the computer to perform an operation. The lesser time it takes to perform an operation, the better is the time complexity. Since there are more than one way to solve a problem, time complexity generally (along with other parameters) is a good way to select the way in which the problem should be solved. The time complexity of any task is dependent on the input size. For instances, if it is independent of the input size, the solution is optimal. Let us consider 2-time complexity examples.

1. The time taken by a Python program to access an element at any given position in the list is instant. It suggests that the time taken to access any large index position in the list is also instant.
2. We are working with a Python list of integer data types, and we need to calculate the sum of the integers in the list. We can say that as the number of integers in the list increases the time taken for the computation of the sum of the integers also increases.

What is Time Complexity of an Algorithm?

In computer programming, an algorithm is a specific set of detailed instructions that are usually carried out by a computer to solve a class of problems or carry out a particular activity. The way the instructions are defined for a given task can vary. To accomplish the same objective, a particular set of instructions might be defined in a variety of ways. Once an algorithm is constructed, we can measure how long it takes to run it by running it on different test inputs and calculating how long it takes for each input. This operation helps us to determine the time complexity of an algorithm. The term "time complexity" describes an algorithm's computational efficiency and shows how the algorithm's runtime grows as the size of the input rises. As per the time complexity definition, it helps in understanding the algorithm's behaviour as input size increases.

The performance of our applications with big data sets is the main worry of computer programmers. To get an understanding of the approximate time it takes for an algorithm to run is measured using the Big-O notation. It measures the time complexity of the algorithm and provides an upper bound to how much time it takes for an algorithm to provide the expected outcome given any input of any size. The Big-O notation also helps in comparing two or more algorithms for time complexity and chose to efficient one. An O(1) complexity signifies that the algorithm’s execution speed is independent of the input size. An O(n) complexity suggests that the algorithm’s execution time grows linearly to the input size. An algorithm having O(2n) time complexity will have an execution time twice to the algorithm having O(n) time complexity.

Let us understand this using source code. We will consider the same example as discussed in the previous section.

Time Complexity: O(1) – Constant

```# GET ITEM AT 'X' INDEX POSITION
def get_item(arr: list, x: int) -> int:
# RETURN THE ELEMENT AT X INDEX POSITION FROM THE LIST
return arr[x] ```

Regardless of the size of the array, this method always returns the element that is located at index location 'x'. Since the runtime is independent of the input size, its time complexity is constant. The function will take the same amount of time to extract the desired element from the array regardless of how many entries it contains.

Time Complexity: O(n) – Linear

```# ADD ITEM IN THE LIST
# INITIALIZE THE EMPTY VARIABLE WITH 0
sum_ = 0
# ITERATE OVER THE ARRAY TO ACCESS EACH ITEM
for item in arr:
# ADD THE NEWY ACCESSED ITEM TO THE ARIGINAL VOLUME COUNTER.
sum_ += item
return sum_ ```

In the above function, all of the array's items are added. The function iterates through each element once as the array size increases, producing a linear relationship between the input size and the processing time.

Time Complexity of Searching algorithms

1. Linear Search

The most basic type of search algorithm is linear search. When using linear search, we iterate through every entry in a list or array until we either locate the item we're looking for or get to the end of the list. As a result, the search takes longer to complete the more items there are in the list. As a result, the linear search algorithm's time complexity is O(n). In the best scenario, the item would be found at the beginning of the list by the search process. In the worst scenario, the item might not be on the list or it might be toward the end. The algorithm traversed through the list from beginning to end in both of these instances.

The Python program to perform linear search is provided below.

```# LINEAR SEARCH IN PYTHON
def linear_search(search_list: list, item: str) -> bool:
# ITERATE FROM 0 TO N-1 INDEX IN THE LIST
for i in range(len(search_list)):
# RETURN TRUE IF THE CURRENT ITEM MATCHES THE SEARCH ITEM
if search_list[i] == item: return True
# RETURN FALSE IF THE ITEM IS NOT FOUND IN THE LIST
return False ```

Here, the function ‘linear_search’ traverses through the list, item by item, and returns ‘True’ if it finds a match indicating that the item is present in the list, else ‘False’ indicating the item is not present in the list. You can analyse the time complexity in python for the above program by providing multiple instances of different input sizes.

2. Binary Search

The average time to search through a list of items for a specific item is O(n), where n is the total number of items in the list. In binary search, the list is sorted first, then a divide-and-conquer strategy can be used to search for an item within the list in O(log n) time. The item in the centre of the sequence is the first thing the binary search algorithm looks for. If it's not there, the binary search algorithm knows where to look—on the left or right side of the sequence—because the list has been sorted. Because it can eliminate half of the remaining elements in each phase, binary search is quite effective. When compared to linear search, this makes it much faster for large datasets. Binary search, however, needs the list to be sorted before using it.

```# BINARY SEARCH IN PYTHON
def binary_search(search_list: list, x: str, low: int, high: int) -> bool:
# RETURN FALSE IF THE MATCH IS NOT FOUND IN THE LIST
if low > high:
return False
else:
# CALCULATE THE MID POSITION OF THE INDEX
mid = low + (high - low)//2
# IF FOUND AT MID POSITION THEN RETURN TRUE
if search_list[mid] == x: return True
# SEARCH THE ITEM IN THE LEFT SIDE OF THE LIST
elif search_list[mid] > x: return binary_search(search_list, x, low, mid - 1)
# SEARCH THE ITEM IN THE RIGHT SIDE OF THE LIST
else: return binary_search(search_list, x, mid + 1, high) ```

Here, the function ‘binary_search’ takes in four compulsory input parameters – search list, the search term, the lower bound index, upper bound index. The lower and upper bound indexes are 0 and one minus the length of the list respectively. If the item is found at the mid position, then the function returns false, otherwise, the item will be searched in the left or the right side after a comparison of the value at the mid position with the search term. In case the lower bound index exceeds the upper bound index, the function returns false, indicating, the item is not present in the list. To learn more about time complexity using python,  python programming and build more search algorithms, you can look forward to learn Python Programming which  is designed to assist you in mastering the concepts of Python and its libraries like SciPy, Matplotlib, Scikit-Learn, Pandas, NumPy, Lambda functions, and Web Scraping.

Why is Time Complexity a Function of its Input Size?

Consider you are asked to manually count the number of people standing in a crowd. The time taken for you to calculate 100 people and 10,000 people is different. Similarly, when we are writing an algorithm, we might have a different approach of solving it. These approaches can work fine with small amount of data points. As the number of data points increase, the algorithm can start to consume more time in providing the desired results. However, there can be other approaches in which the time consumption would be lesser than the other ones. We may predict the algorithm's performance as the input size increases by creating a measure that links time complexity to input size. This approximation helps to analyse various algorithms and evaluate their effectiveness for various input data sizes. Understanding the behaviour and scalability of an algorithm can be made easier by expressing different types of time complexity as a function of input size rather than relying solely on hardware or constant variables. It simplifies the process of selecting the best algorithms for a given problem size and optimizes algorithm selection for practical uses.

To understand the relationship between time complexity and the input size, let us consider an example where we are comparing an array with itself to find the items that have been repeated in the array.

```# CHECK ITEMS REPEATED IN THE LIST
def get_repitition(arr: list) -> list:
# DECLARE AN EMPTY LIST FOR STORING THE ITEMS
response = []
# ITERATE THROUGH THE LIST TO PICK THE FIRST ITEM FOR COMPARISON
for i in range(len(arr)):
# ITERATE THROUGH THE LIST TO PICK SECOND ITEM FOR COMPARISON
for j in range(len(arr)):
# SKIP THE ITERATION IF THE INDEXES ARE SAME
if i == j: continue
# CHECK IF THE ITEMS ARE SAME AND APPEND IT TO THE RESPONSE LIST
if arr[i] == arr[j]: response.append(arr[i])
# RETURN THE RESPONSE LIST
return response```

It's awful when we have a loop inside a loop, and we do nested iteration since the time complexity is quadratic. There will be a total of n-square iterations since the outer loop will run n times and the inner loop will run n times for each iteration of the outer loop. This is what happens in the above code. The outer loop will provide a place holder for first item for comparison, while the inner loop will provide a place holder for the second item for comparison. The Big-O notation is therefore, given by O(n2).

In the above time complexity graph, we can see some of the different time complexities that are involved in an operation and how they perform. The complexities mentioned in the red zone, such as, O(n!), O(2n), and O(n2), have horrible speed especially when the input size gets larger. The time complexity of O(n.log n) is considered to be fair or below average depending on the operation we are working on. The linear time complexity O(n) is considered to be fairly good whereas O(log n) and O(1) are the best time complexities that an operation can achieve.

Conclusion

In this article, we learnt about what is time complexity and its significance. We also had a brief look at Big-O notation and how it helps us to compare two different algorithms in terms of their execution speed. We have demonstrated Python examples for better understanding. However, one can refer to any programming language for implementing the algorithms. If the set of instructions are the same, then each of the language will implement the algorithm with the same time complexity. To know more on how we can select the right algorithms and refine the algorithms with poor complexities, you can head over to Java Programming for beginners course, where you can learn from leading Java experts who have trained more than 400,000 professionals. With more than 90 hours of hands-on training and enough study resources, the course is the best place to start with.

1What are the time complexity examples?

Here are the time complexity examples such as constant time complexity, linear time complexity, logarithmic time complexity, linearities time complexity, quadratic time complexity, exponential time complexity

2Why do we use time complexity?

We used time complexity to calculate how long it takes to execute each code statement in an algorithm.

3What affects time complexity?

The time complexity of an algorithm is required to execute a code that depends on other factors like a programming language, operating software, processing power, etc.

4What is the time complexity formula?

T(n) = n - 1 is the time complexity as measured by the number of comparisons. The time complexity for linear search, for example, can be represented as O(n) and the time complexity for binary search as O(log n).

Ramulu Enugurthi

Blog Author

Ramulu Enugurthi, a distinguished computer science expert with an M.Tech from IIT Madras, brings over 15 years of software development excellence. Their versatile career spans gaming, fintech, e-commerce, fashion commerce, mobility, and edtech, showcasing adaptability in multifaceted domains. Proficient in building distributed and microservices architectures, Ramulu is renowned for tackling modern tech challenges innovatively. Beyond technical prowess, he is a mentor, sharing invaluable insights with the next generation of developers. Ramulu's journey of growth, innovation, and unwavering commitment to excellence continues to inspire aspiring technologists.