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HomeBlogProgrammingTime Complexity: Significance, Types, Algorithms
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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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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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.
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:
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).
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.
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:
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.
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.
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.
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.
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:
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:
Second Move:
Third Move:
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).
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.
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.
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.
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.
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.
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, 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.
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 def add_item(arr): # 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.
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.
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.
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.
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.
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
We used time complexity to calculate how long it takes to execute each code statement in an algorithm.
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.
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).
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