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- Write a pseudocode for the following problem statement and clearly mention the time complexity in big O notation. Given an array of N integers, determine whether the sequence is in strictly increasing order (i.e. each number is greaterthan the last). Return True if condition is satisfied, False otherwise.1. Big-O notation. We have learnt big-O notation to compare the growth rates of functions, this exercise helps you to better understand its definition and properties.(a) Suppose n is the input size, we have the following commonly seenfunctions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n^2, f6(n) = 2^n, f7(n) = n!, f8(n) = n^n. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n^2 < 2^n < n! < n^n. Provethis is true.[Hint: You are expected to prove the following asymptotics by using the definition of big-O notation: 1 = O(log n), log n = O(n), n = O(n log n), n log n = O(n^2), n^2 = O(2^n), 2^n = O(n!), n! = O(n^n). Note: Chap 3.2 of our textbook provides some math facts in case you need.]What is the leading term for the following expressions and specify the lowest Big O Complexity for each algorithm?
- Below is a list of functions that commonly appear in complexity analyzes as a function of the size n of the problem. Sort the functions in ascending order of growth rate, that is, the slowest growing one is 1, and so on. Note that the functions are described in the notation O(.), which represents the cost of the algorithm as a function of the predominant term of the cost expression.#4. Euler's totient function, also known as phi-function ϕ(n),counts the number of integers between 1 and n inclusive,which are coprime to n.(Two numbers are coprime if their greatest common divisor (GCD) equals 1)."""def euler_totient(n): """Euler's totient function or Phi function. Time Complexity: O(sqrt(n)).""" result = n for i in range(2, int(n ** 0.5) + 1): if n % i == 0: while n % i == 0: n //= i.CODING QUESTION. Euler's totient function, also known as phi-function ϕ(n),counts the number of integers between 1 and n inclusive,which are coprime to n.(Two numbers are coprime if their greatest common divisor (GCD) equals 1)."""def euler_totient(n): """Euler's totient function or Phi function. Time Complexity: O(sqrt(n)).""" result = n.
- Exercise 2. Give a recursive definition for the factorial operation k! n! for n ≥ 1. (remember that 1! = 0! = 1) Provide an algorithm in pseudo code to evaluate k! n! as one function Provide an algorithm in pseudo code to evaluate k! n! as three functions Evaluate the complexity of the algorithm at point 2 Evaluate the complexity of the algorithm at point 3Question 2 Using the incomplete programming code given, complete the code using dynamic programming with memory function, to reproduce the results in the following Table 1. (C++) #include<iostream>using namespace std; // max knapsack capacity // *** WRITE YOUR CODE HERE ***// num of items // *** WRITE YOUR CODE HERE ***// weight of each item // *** WRITE YOUR CODE HERE ***// value of each item // *** WRITE YOUR CODE HERE ***// variable for dynamic programming matrix // *** WRITE YOUR CODE HERE *** //==========================================// Dynamic programming function: recursive// ========================================= // ALGORITHM F(i,j) // int value // if F[i,j] is not filled yet (-1): // (start with j = W, i = n) // if capacity j < current item's weight w[i]: // value = recall F(i-1, j) // else: // we can include current item,…Analyze the following codes for Time and space complexity. Determine Big O for the following code fragments in the average case. Assume that all variables are of type int. (i) sum = 0; if (EVEN(n)) for (i=0; i<n; i++) sum++; else sum = sum + n; sum1 = 0;
- Problem 1: Euclid’s algorithm (or the Euclidean algorithm) is an algorithm that computes thegreatest common divisor, denoted by gcd, of two integers. Below are the original versions ofEuclid’s algorithm that uses repeated subtraction and another one that uses the remainder.int gcd_sub(int a, int b){ if (!a) return b; while (b) if (a > b) a = a – b; else b = b – a; return a;}int gcd_rem(int a, int b){ int t; while (b) { t = b; b = a % b; a = t; } return a;}1. Trace each of the above algorithm using specific values for a and b.2. Compare both algorithms.Problem 2: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder…Answer the given question with a proper explanation and step-by-step solution. C++ 11.12 LAB: Fibonacci sequence (recursion) The Fibonacci sequence begins with 0 and then 1 follows. All subsequent values are the sum of the previous two, for example: 0, 1, 1, 2, 3, 5, 8, 13. Complete the Fibonacci() function, which takes in an index, n, and returns the nth value in the sequence. Any negative index values should return -1. Ex: If the input is: 7 the output is: Fibonacci(7) is 13 Note: Use recursion and DO NOT use any loops. main.cpp #include <iostream>using namespace std; int Fibonacci(int n) {/* Type your code here. */ } int main() {int startNum;cin >> startNum;cout << "Fibonacci(" << startNum << ") is " << Fibonacci(startNum) << endl;return 0;}Assume that you were given N cents (N is an integer) and you were asked to break up the N cents into coins consisting of 1 cent, 2 cents and 5 cents. Write a dynamicprogramming based recursive algorithm, which returns the smallest (optimal) number of coins needed to solve this problem. For example, if your algorithm is called A, and N = 13, then A(N) = A(13) returns 4, since 5+5+2+1 = 13 used the smallest (optimal) number of coins. In contrast, 5+5+1+1+1 is not an optimal answer.