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Time Complexity: Significance, Types, Algorithms
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In Computer Science, there are various ways of solving problems. These methods may imply different times, computational power, or any other metric you choose, so we must compare the efficiency of various approaches to select the best one. Now we all know that computers can solve problems based on algorithms. “Algorithms are the instructions that tell the computer how to execute the particular programs”. Let us consider the problem of preparing an omelet. To prepare an omelet, we follow the steps given below:
- Get the frying pan.
- Get the oil.
- Do we have oil?
- If yes, put it in the pan
- If not, do we want to buy oil?
- If yes, then go out and buy.
- If not, we can terminate.
- Do we have oil?
- Turn on the stove, etc.
The algorithm and flow chart is one of the two types of tools that explain the process of the program.
Here is an example of converting temperature from Fahrenheit to Celsius.
Algorithm:
- Step 1: Read the temperature from Fahrenheit.
- Step 2: Calculate the temperature using the Formula, i.e., C = 5/9*(F-32)
- Step 3: Print the C Value.
We will now understand through the flow diagram.
By looking at this flow diagram, we understood that diagram is a pictorial representation of an algorithm. With this, an algorithm can express and analyze the flow chart.
An Algorithm is the unambiguous step-by-step instructions to solve a given problem.
Fortunately, there are methods for accomplishing this, and we don't need to watch the algorithm in action to determine whether it can complete the task quickly or if it will collapse under the weight of its input. When considering the complexity of an algorithm, we shouldn't be concerned with the exact number of operations performed; rather, we should be concerned with how the number of operations relates to the size of the problem. Consider this: if the problem size doubles, does the number of operations remain constant? Do they multiply? Do they grow in any other way? We need to measure the time complexity of algorithms in order to answer these questions.
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.
4. Quadratic Time Complexity: O(n²)
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:
- 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.
- Therefore, it replaces 23 with 12, the lowest value in the array. After, it presents the first index in the sorted list.
Second Move:
- For the second position, 56 are present. Again, it will be traversed with the other array list in a sequential manner.
- 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:
- 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.
- 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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Conclusion
In this article, we explored the fundamental concepts of Time complexity and the importance of using it in the algorithms. We also understood the various types of time complexities used for different types of functions and how to assign the order of notation for any algorithm based on the cost function and the number of times the statement is defined to run. We know that Notation and Time Complexity play an important role in understanding the algorithm properly.
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Frequently Asked Questions (FAQs)
1. What 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
2. Why do we use time complexity?
We used time complexity to calculate how long it takes to execute each code statement in an algorithm.
3. What 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.
4. What 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).
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