Linear Time Complexity O(n)
The Time Complexity of a loop is considered as O(n) if the loop variables are incremented/decremented by a constant amount. For example following functions have O(n) time complexity. Linear time complexity, denoted as O(n), is a measure of the growth of the running time of an algorithm proportional to the size of the input. In an O(n) algorithm, the running time increases linearly with the size of the input. For example, searching for an element in an unsorted array or iterating through an array and performing a constant amount of work for each element would be O(n) operations. In simple words, for an input of size n, the algorithm takes n steps to complete the operation.
C++
// Here c is a positive integer constantfor (int i = 1; i <= n; i = i + c) { // some O(1) expressions}for (int i = n; i > 0; i = i - c) { // some O(1) expressions}// This code is contributed by Kshitij |
C
// Here c is a positive integer constantfor (int i = 1; i <= n; i += c) { // some O(1) expressions}for (int i = n; i > 0; i -= c) { // some O(1) expressions} |
Java
// Here c is a positive integer constantfor (int i = 1; i <= n; i += c) { // some O(1) expressions} for (int i = n; i > 0; i -= c) { // some O(1) expressions}// This code is contributed by Utkarsh |
C#
for (int i = 1; i <= n; i = i + c) { // some O(1) expressions // O(1) expressions could be computations, assignments, // or other constant time operations}// Second loop: Decrementing by 'c' from n to 1for (int i = n; i > 0; i = i - c) { // some O(1) expressions // O(1) expressions could be computations, assignments, // or other constant time operations} |
Javascript
// Here c is a positive integer constantfor (var i = 1; i <= n; i += c) { // some O(1) expressions}for (var i = n; i > 0; i -= c) { // some O(1) expressions} |
Python3
# Here c is a positive integer constantfor i in range(1, n+1, c): # some O(1) expressionsfor i in range(n, 0, -c): # some O(1) expressions # This code is contributed by Pushpesh Raj |
How to Analyse Loops for Complexity Analysis of Algorithms
We have discussed Asymptotic Analysis, Worst, Average and Best Cases and Asymptotic Notations in previous posts. In this post, an analysis of iterative programs with simple examples is discussed.
The analysis of loops for the complexity analysis of algorithms involves finding the number of operations performed by a loop as a function of the input size. This is usually done by determining the number of iterations of the loop and the number of operations performed in each iteration.
Here are the general steps to analyze loops for complexity analysis:
Determine the number of iterations of the loop. This is usually done by analyzing the loop control variables and the loop termination condition.
Determine the number of operations performed in each iteration of the loop. This can include both arithmetic operations and data access operations, such as array accesses or memory accesses.
Express the total number of operations performed by the loop as a function of the input size. This may involve using mathematical expressions or finding a closed-form expression for the number of operations performed by the loop.
Determine the order of growth of the expression for the number of operations performed by the loop. This can be done by using techniques such as big O notation or by finding the dominant term and ignoring lower-order terms.
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