Amortized Analysis is used for algorithms where an occasional operation is very slow, but most other operations are faster. In Amortized Analysis, we analyze a sequence of operations and guarantee a worst-case average time that is lower than the worst-case time of a particularly expensive operation.…

The analysis of the complexity of a recurrence relation involves finding the asymptotic upper bound on the running time of a recursive algorithm. This is usually done by finding a closed-form expression for the number of operations performed by the algorithm as a function of the input size, and then…

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…

Prerequisite: Analysis of Algorithms 1. What is the time, and space complexity of the following code: CPP int a = 0, b = 0; for (i = 0; i < N; i++) { a = a + rand(); } for (j = 0; j < M; j++) { b = b + rand(); } Java int a = 0, b = 0; for (i = 0; i < N; i++) { a = a + Math.random(); } for (j = 0; j…

Prerequisite: Analysis of Algorithms | Big-O analysis In the previous article, the analysis of the algorithm using Big O asymptotic notation is discussed. In this article, some examples are discussed to illustrate the Big O time complexity notation and also learn how to compute the time complexity…

1. Big O notation (O): It defines an upper bound on order of growth of time taken by an algorithm or code with input size. Mathematically, if f(n) describes the running time of an algorithm; f(n) is O(g(n)) if there exist positive constant C and n0 such that, 0 <= f(n) <= Cg(n) for all n >= n0 n =…

The term Space Complexity is misused for Auxiliary Space at many places. Following are the correct definitions of Auxiliary Space and Space Complexity. Auxiliary Space is the extra space or temporary space used by an algorithm. The space Complexity of an algorithm is the total space taken by the…

A lot of students get confused while understanding the concept of time complexity, but in this article, we will explain it with a very simple example. Q. Imagine a classroom of 100 students in which you gave your pen to one person. You have to find that pen without knowing to whom you gave it. Here…

In the analysis of algorithms, asymptotic notations are used to evaluate the performance of an algorithm, in its best cases and worst cases. This article will discuss Big-Omega Notation represented by a Greek letter (Ω). Table of Content What is Big-Omega Ω Notation? Big-Omega Ω Notation, is a way…

In the analysis of algorithms, asymptotic notations are used to evaluate the performance of an algorithm by providing an exact order of growth. This article will discuss Big - Theta notations represented by a Greek letter (Θ). Definition: Let g and f be the function from the set of natural numbers…

Big O notation is a powerful tool used in computer science to describe the time complexity or space complexity of algorithms. Big-O is a way to express the upper bound of an algorithm’s time or space complexity. Describes the asymptotic behavior (order of growth of time or space in terms of input…

Given two algorithms for a task, how do we find out which one is better? One naive way of doing this is - to implement both the algorithms and run the two programs on your computer for different inputs and see which one takes less time. There are many problems with this approach for the analysis of…

In the previous post, we discussed how Asymptotic analysis overcomes the problems of the naive way of analyzing algorithms. Now let us learn about What is Worst, Average, and Best cases of an algorithm: 1. Worst Case Analysis (Mostly used) In the worst-case analysis, we calculate the upper bound on…

Let f(n) and g(n) be the time taken by two algorithms where n >= 9 and f(n) and g(n) are also greater than equal to 0. A function f(n) is said to be growing faster than g(n) if g(n)/f(n) for n tends to infinity is 0 (or f(n)/g(n) for n tends to infinity is infinity). Example 1: f(n) = 1000, g(n) = n…

Why is Performance of Algorithms Important ? There are many important things that should be taken care of, like user-friendliness, modularity, security, maintainability, etc. Why worry about performance? The answer to this is simple, we can have all the above things only if we have performance. So…

Last Updated : DiscussComments Question 1 What is time complexity of fun()? C++ int fun(int n) { int count = 0; for (int i = n; i > 0; i /= 2) for (int j = 0; j < i; j++) count += 1; return count; } C int fun(int n) { int count = 0; for (int i = n; i > 0; i /= 2) for (int j = 0; j < i; j++) count +=…

Try it on GfG Practice Given an array arr[] of non-negative numbers, the task is to find GCD of all the array elements. In a previous post we find GCD of two number. Examples: Input: arr[] = [1, 2, 3] Output: 1 Input: arr[] = [2, 4, 6, 8] Output: 2 Using Recursive GCD The GCD of three or more…

Two numbers A and B are said to be Co-Prime or mutually prime if the Greatest Common Divisor of them is 1. You have been given two numbers A and B, find if they are Co-prime or not. Examples : Input : 2 3 Output : Co-Prime Input : 4 8 Output : Not Co-Prime The idea is simple, we find GCD of two…

Try it on GfG Practice Given two integer arrays a[] and b[] containing two integers each representing the numerator and denominator of a fraction respectively. The task is to find the sum of the two fractions and return the numerator and denominator of the result. Examples : Input: a = [1, 2] , b =…

Try it on GfG Practice Given two positive integers a and b. Find the Least Common Multiple (LCM) of a and b. LCM of two numbers is the smallest number which can be divided by both numbers. Input : a = 10, b = 5 Output : 10 Explanation : 10 is the smallest number divisible by both 10 and 5 Input : a…

Try it on GfG Practice Given an integer n, find whether the number is Palindrome or not. A number is a Palindrome if it remains the same when its digits are reversed. Examples: Input: n = 12321 Output: Yes Explanation: 12321 is a Palindrome number because after reversing its digits, the number…

Try it on GfG Practice Given three numbers a, b and k, find k-th digit in ab from right side Examples: Input : a = 3, b = 3, k = 1 Output : 7 Explanation: 3^3 = 27 for k = 1. First digit is 7 in 27 Input : a = 5, b = 2, k = 2 Output : 2 Explanation: 5^2 = 25 for k = 2. First digit is 2 in 25 The…

Try it on GfG Practice Given an Integer n, find the reverse of its digits. Examples: Input: n = 122 Output: 221 Explanation: By reversing the digits of number, number will change into 221. Input: n = 200 Output: 2 Explanation: By reversing the digits of number, number will change into 2. Input: n =…

Try it on GfG Practice Given a number n, find the sum of its digits. Examples : Input: n = 687 Output: 21 Explanation: The sum of its digits are: 6 + 8 + 7 = 21 Input: n = 12 Output: 3 Explanation: The sum of its digits are: 1 + 2 = 3 Table of Content [Approach 1] Digit Extraction - O(log10n) Time…

Given a number n, the task is to return the count of digits in this number. Example: Input: n = 1567 Output: 4 Explanation: There are 4 digits in 1567, which are 1, 5, 6 and 7. Input: n = 255 Output: 3 Explanation: The are 3 digits in 256, which are 2, 5 and 5. Input: n = 58964 Output: 5…

Try it on GfG Practice Given a positive integer n. The task is to find the sum of the sum of first n natural number. Examples: Input: n = 3 Output: 10 Explanation: Sum of first natural number: 1 Sum of first and second natural number: 1 + 2 = 3 Sum of first, second and third natural number = 1 + 2 +…

Try it on GfG Practice A number is termed a triangular number if we can represent it in the form of a triangular grid of points such that the points form an equilateral triangle and each row contains as many points as the row number, i.e., the first row has one point, second row has two points,…

Try it on GfG Practice Given three integers — the first term a, common ratio r, and position n — the objective is to find the n-th term of a geometric progression. Note: Since the result can be large, return the answer modulo 10^9 + 7. Examples: Input: a = 2, r = 2, n = 4 Output: 16 Explanation: The…

Try it on GfG Practice Given first term (a), common difference (d) and a integer N of the Arithmetic Progression series, the task is to find Nthterm of the series. Examples : Input : a = 2 d = 1 N = 5 Output : 6 The 5th term of the series is 6. The terms are 2, 3, 4, 5, 6, .... Input : a = 5 d = 2 N…

Try it on GfG Practice Given a positive integer n, we have to find the sum of squares of first n natural numbers. Examples : Input : n = 2 Output: 5 Explanation: 1^2+2^2 = 5 Input : n = 8 Output: 204 Explanation : 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204 [Naive Approach] - Adding One By…

Try it on GfG Practice Given a positive integer n, find the sum of the first n natural numbers. Examples : Input: n = 3 Output: 6 Explanation: 1 + 2 + 3 = 6 Input: n = 5 Output: 15 Explanation: 1 + 2 + 3 + 4 + 5 = 15 Table of Content [Naive Approach] Using Loop - O(n) Time and O(1) Space Calculate…

Given a large number n, find if the number is divisible by 29. Examples : Input : 363927598 Output : No Input : 292929002929 Output : Yes A quick solution to check if a number is divisible by 29 or not is to add 3 times of last digit to rest number and repeat this process until number comes 2 digit.…

Given a number s represented as a string, determine whether the integer it represents is divisible by 13 or not. Examples : Input: s = "2911285" Output: true Explanation: 2911285 ÷ 13 = 223945, which is a whole number with no remainder. Input: s = "920" Output: false Explanation: 920 ÷ 13 =…

Given a number in the form of string s, Check if the number is divisible by 11 or not. The input number may be large and it may not be possible to store it even if we use long long int. Examples: Input: s = "76945" Output: true Explanation: s when divided by 11 gives 0 as remainder. Input: s =…

Try it on GfG Practice Given a number, the task is to check if a number is divisible by 4 or not. The input number may be large and it may not be possible to store even if we use long long int. Examples: Input : n = 1124 Output : Yes Input : n = 1234567589333862 Output : No Input : n =…

Given a number, the task is that we divide number by 3. The input number may be large and it may not be possible to store even if we use long long int. Examples: Input : n = 769452 Output : Yes Input : n = 123456758933312 Output : No Input : n = 3635883959606670431112222 Output : Yes Since input…

Try it on GfG Practice Given a number n, check if it is a perfect square or not. Examples : Input : n = 36 Output : Yes Input : n = 2500 Output : Yes Explanation: 2500 is a perfect square of 50 Input : n = 8 Output : No Table of Content Using sqrt() Take the floor()ed square root of the…

Try it on GfG Practice Given a number x, determine whether the given number is Armstrong's number or not. A positive integer of n digits is called an Armstrong number of order n (order is the number of digits) if abcd... = pow(a,n) + pow(b,n) + pow(c,n) + pow(d,n) + .... Here a, b, c and d are…

Try it on GfG Practice Given a number n, print all numbers in the range from 1 to n having exactly 3 divisors. Examples: Input: n = 16 Output: 4 9 Explanation: 4 and 9 have exactly three divisors. Input: n = 49 Output: 4 9 25 49 Explanation: 4, 9, 25 and 49 have exactly three divisors. [Naive…

Try it on GfG Practice Given two positive numbers x and y, check if y is a power of x or not. Examples : Input: x = 10, y = 1 Output: True x^0 = 1 Input: x = 10, y = 1000 Output: True x^3 = 1 Input: x = 10, y = 1001 Output: False Table of Content [Naive Approach] Repeated Multiplication Method This…