Design and Analysis of Algorithms

Design and Analysis of Algorithms is a fundamental aspect of computer science that involves creating efficient solutions to computational problems and evaluating their performance. DSA focuses on designing algorithms that effectively address specific challenges and analyzing their efficiency in terms of time and space complexity .

Complete Guide On Complexity Analysis

Table of Content

• What is meant by Algorithm Analysis?
• Why Analysis of Algorithms is important?
• Types of Algorithm Analysis
• Basics on Analysis of Algorithms
• Asymptotic Notations
• Some Advance topics
• Complexity Proofs

Basics on Analysis of Algorithms:

• What is algorithm and why analysis of it is important?
• Asymptotic Notation and Analysis (Based on input size) in Complexity Analysis of Algorithms
• Worst, Average and Best Case Analysis of Algorithms
• Types of Asymptotic Notations in Complexity Analysis of Algorithms
• How to Analyse Loops for Complexity Analysis of Algorithms
• How to analyse Complexity of Recurrence Relation
• Introduction to Amortized Analysis

Asymptotic Notations:

• Analysis of Algorithms | Big-O analysis
• Difference between Big Oh, Big Omega and Big Theta
• Examples of Big-O analysis
• Difference between big O notations and tilde
• Analysis of Algorithms | Big – Ω (Big- Omega) Notation
• Analysis of Algorithms | Big – Θ (Big Theta) Notation

Some Advance topics:

• Types of Complexity Classes | P, NP, CoNP, NP hard and NP complete
• Can Run Time Complexity of a comparison-based sorting algorithm be less than N logN?
• Why does accessing an Array element take O(1) time?
• What is the time efficiency of the push(), pop(), isEmpty() and peek() operations of Stacks?

Complexity Proofs:

• Proof that Clique Decision problem is NP-Complete
• Proof that Independent Set in Graph theory is NP Complete
• Prove that a problem consisting of Clique and Independent Set is NP Complete
• Prove that Dense Subgraph is NP Complete by Generalisation
• Prove that Sparse Graph is NP-Complete

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Asymptotic Analysis is defined as the big idea that handles the above issues in analyzing algorithms. In Asymptotic Analysis, we evaluate the performance of an algorithm in terms of input size (we don't measure the actual running time). We calculate, how the time (or space) taken by an algorithm increases with the input size. Asymptotic notation is

What is the time complexity of fun()? C/C++ Code int fun(int n) < int count = 0; for (int i = 0; i 0; j--) count = count + 1; return count; >(A) Theta (n) (B) Theta (n2) (C) Theta (n*log(n)) (D) Theta (n*(log(n*log(n)))) Answer: (B)Explanation: The time complexity can be calculated by counting the number of times the expression "count = count + 1;

The recurrence relation capturing the optimal time of the Tower of Hanoi problem with n discs is. (GATE CS 2012) (A) T(n) = 2T(n – 2) + 2 (B) T(n) = 2T(n – 1) + n (C) T(n) = 2T(n/2) + 1 (D) T(n) = 2T(n – 1) + 1 Answer: (D)Explanation: Following are the steps to follow to solve Tower of Hanoi problem recursively. Let the three pegs be A, B and C. Th

O( n2 ) is the worst case time complexity, so among the given options it can represent :- (A) O( n ) (B) O( 1 ) (C) O ( nlogn ) (D) All of the above Answer: (D) Explanation: O( n2 ) is the worst case time complexity, so, if the time complexity is O( n2 ), they all can be represented by it. Quiz of this Question Please comment below if you find anyt

Which of the following is not O(n2)? (A) (15) * n2 (B) n1.98 (C) n3/(sqrt(n)) (D) (20) * n2 Answer: (C) Explanation: The order of growth of option c is n2.5 which is higher than n2. Quiz of this Question Please comment below if you find anything wrong in the above post

Which of the given options provides the increasing order of asymptotic complexity of functions f1, f2, f3, and f4? f1(n) = 2n f2(n) = n(3/2) f3(n) = n*log(n) f4(n) = nlog(n) (A) f3, f2, f4, f1 (B) f3, f2, f1, f4 (C) f2, f3, f1, f4 (D) f2, f3, f4, f1 Answer: (A)Explanation: f1(n) = 2^n f2(n) = n^(3/2) f3(n) = n*log(n) f4(n) = n^log(n) Except for f3,

Consider the following program fragment for reversing the digits in a given integer to obtain a new integer. Let n = D1D2…Dm int n, rev; rev = 0; while (n > 0) < rev = rev*10 + n%10; n = n/10; >The loop invariant condition at the end of the ith iteration is: (GATE CS 2004) (A) n = D1D2….Dm-i and rev = DmDm-1…Dm-i+1 (B) n = Dm-i+1…Dm-1Dm and rev

What is the time complexity of the below function? C/C++ Code ) < int i = 0, j = 0; for (; i (A) O(n) (B) O(n2) (C) O(n*log(n)) (D) O(n*log(n)2) Answer: (A)Explanation: At first look, the time complexity seems to be O(n2) due to two loops. But, please note that the variable j is not initialized for each value of variable i. So, the inner loop runs

In a competition, four different functions are observed. All the functions use a single for loop and within the for loop, same set of statements are executed. Consider the following for loops: C/C++ Code A) for(i = 0; i If n is the size of input(positive), which function is most efficient(if the task to be performed is not an issue)? (A) A (B) B (C

The following statement is valid. log(n!) = [Tex]\\theta [/Tex](n log n). (A) True (B) False Answer: (A)Explanation: Order of growth of \\log n! and n\\log n is the same for large values of , i.e., \\theta (\\log n!) = \\theta (n\\log n) . So time complexity of fun() is \\theta (n\\log n) . The expression \\theta (\\log n!) = \\theta (n\\log n) can

What does it mean when we say that an algorithm X is asymptotically more efficient than Y? (A) X will be a better choice for all inputs (B) X will be a better choice for all inputs except possibly small inputs (C) X will be a better choice for all inputs except possibly large inputs (D) Y will be a better choice for small inputs Answer: (B)Explanat

What is the time complexity of Floyd–Warshall algorithm to calculate all pair shortest path in a graph with n vertices? (A) O(n2log(n)) (B) Theta(n2log(n)) (C) Theta(n4) (D) Theta(n3) Answer: (D)Explanation: Floyd–Warshall algorithm uses three nested loops to calculate all pairs shortest path. So, the time complexity is Theta(n3).Please read here f

Consider the following functions: f(n) = 2n g(n) = n! h(n) = nlog(n) Which of the following statements about the asymptotic behavior of f(n), g(n), and h(n) is true?(A) f(n) = O(g(n)); g(n) = O(h(n))(B) f(n) = [Tex]\\Omega [/Tex](g(n)); g(n) = O(h(n))(C) g(n) = O(f(n)); h(n) = O(f(n))(D) h(n) = O(f(n)); g(n) = [Tex]\\Omega [/Tex](f(n)) (A) A (B) B

Consider the following functions f(n) = 3n^>g(n) = 2^<\log_<2>n>>h(n) = n! Which of the following is true? (GATE CS 2000)(A) h(n) is 0(f(n))(B) h(n) is 0(g(n))(C) g(n) is not 0(f(n))(D) f(n) is 0(g(n)) (A) A (B) B (C) C (D) D Answer: (D)Explanation: g(n) = 2^[Tex](\\sqrt \\log ) [/Tex]= n^[Tex](\\sqrt) [/Tex]f(n) and g(n) 1 min read Algorithms | Analysis of Algorithms | Question 17

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Consider the following function, int unknown(int n) < int i, j, k = 0; for (i = n/2; i <= n; i++) for (j = 2; j <= n; j = j * 2) k = k + n/2; return k; >What is the time complexity of the function? (GATE CS 2013) (A) n^2 (B) n logn (C) n^3 (D) n^3 logn Answer: (B)Explanation: In the below explanation, \'^\' is used to represent exponent: The

Consider the following two functions. What are time complexities of the functions? C/C++ Code int fun1(int n) < if (n C/C++ Code int fun2(int n) < if (n (A) O(2^n) for both fun1() and fun2() (B) O(n) for fun1() and O(2^n) for fun2() (C) O(2^n) for fun1() and O(n) for fun2() (D) O(n) for both fun1() and fun2() Answer: (B)Explanation: Time complexity

What is time complexity of fun()? C/C++ Code int fun(int n) < int count = 0; for (int i = n; i >0; i /= 2) for (int j = 0; j (A) O(n2) (B) O(n*log(n)) (C) O(n) (D) O(n*log(n*Log(n))) Answer: (B) Explanation: For an input integer n, the outermost loop of fun() is executed log(n) times the innermost statement of fun() is executed following times. n

In the following C function, let n >= m. C/C++ Code int gcd(n, m) < if (n % m == 0) return m; n = n % m; return gcd(m, n); >How many recursive calls are made by this function?(A) [Tex]\\theta [/Tex](log(n))(B) [Tex]\\Omega [/Tex](n)(C) [Tex]\\theta [/Tex](log(log(n)))(D) [Tex]\\theta [/Tex](sqrt(n)) (A) A (B) B (C) C (D) D Answer: (A) Explanati

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In the previous post, we discussed how Asymptotic analysis overcomes the problems of the naive way of analyzing algorithms. But let's take an overview of the asymptotic notation and learn about What is Worst, Average, and Best cases of an algorithm: Popular Notations in Complexity Analysis of Algorithms1. Big-O NotationWe define an algorithm’s wors

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• Algorithms
• DSA
• Algorithms-Analysis of Algorithms