The function PowerRecursive below takes inputs r and n, wherer is any real number and n is a non-negative integer. The output is r". PowerRecursive(r, n) If (n = 0), Return (1) y := PowerRecursive(r, n-1) Return (r·y) The number of atomic operations performed by PowerRecursive does not depend on the value of r. The function T(n) is the number of atomic operations performed by the PowerRecursive on input n. The question below derives the recurrence relation for the time complexity of the algorithm PowerRecursive. What is the number of atomic operations performed by the recursive call in PowerRecursive?
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- How can the following function be simplified so that it has a time complexity of O(n) or faster?For your information, the specifications of the functions are as follows: Using numbers ranging from 0 to 15 (inclusive), create all possible lists which sum up to K and have a length of N. Duplicated numbers are allowed as long as it fulfills the conditions above (this means [0,0,1], [0,1,0] and [1,0,0] are all correct outputs if K=1 and N=3). When instantiated with the list function, the list size of the function should be the number of all lists. For example, given K=23 and N=2, the expected list size is 8. The function must be able to accept N=10 and be finished before 9 seconds. Do not use itertools or external libraries.The following function f uses recursion: def f(n): if n <= 1 return n else return f(n-1) + f(n-2) 5 Let n be a valid input, i.e., a natural number. Which of the following functions returns the same result but without recursion? a) def f(n): a <- 0 b <- 1 if n = 0 return a elsif n = 1 return b else for i in 1..n c <- a + b a <- b b <- c return b f(n): a <- 0 i <- n while i > 0 a <- a + i + (i-1) return a f(n): arr[0] <- 0 arr[1] <- 1 if n <= 1 return arr[n] else for i in 2..n arr[i] <- arr[i-1] + arr[i-2] return arr[n] f(n): arr[0..n] <- [0, ..., n] if n <= 1 return arr[n] else a <- 0 for i in 0..n a <- a + arr[i] return aImplement the quadratic_formula() function. The function takes 3 arguments, a, b, and c, and computes the two results of the quadratic formula: x1=−b+b2−4ac2a x2=−b−b2−4ac2a The quadratic_formula() function returns the tuple (x1, x2). Ex: When a = 1, b = -5, and c = 6, quadratic_formula() returns (3, 2). Code provided in main.py reads a single input line containing values for a, b, and c, separated by spaces. Each input is converted to a float and passed to the quadratic_formula() function. Ex: If the input is: 2 -3 -77 the output is: Solutions to 2x^2 + -3x + -77 = 0 x1 = 7 x2 = -5.50 code: # TODO: Import math module def quadratic_formula(a, b, c):# TODO: Compute the quadratic formula results in variables x1 and x2return (x1, x2) def print_number(number, prefix_str):if float(int(number)) == number:print("{}{:.0f}".format(prefix_str, number))else:print("{}{:.2f}".format(prefix_str, number)) if __name__ == "__main__":input_line = input()split_line = input_line.split(" ")a =…
- implement a function for finding the nth Fibonacci number using the MIPS assembly language. Following gives the definition of the Fibonacci Sequence.Let F be the Fibonacci function:F(0) = 0F(1) = 1For n > 1 F(n) = F(n-1) + F(n-2)Example Output of Execution:Enter the sequence number: 6F(6) = 8 A - Loop ImplementationIn a file named FiboLoop.asm, implement the above function using a loop. B - Recursive ImplementationIn a file named FiboRec.asm, implement the above function using a loop.In C programming Mathematically, given a function f, we recursively define fk(n) as follows: if k = 1, f1(n) = f(n). Otherwise, for k > 1, fk(n) = f(fk-1(n)). Assume that there is an existing function f, which takes in a single integer and returns an integer. Write a recursive function fcomp, which takes in both n and k (k > 0), and returns fk(n). int f(int n);int fcomp(int n, int k){procedure Horner(c, a_(0), a_(1), a_(2),… , a_(n): real numbers)y := a_(n)for i := 1 to ny := y ∗ c + a_(n−i)return y {y = a_(n)c^(n) + a_(n−1)c^(n−1) + ⋯ + a_(1)c + a_(0)}a) Evaluate 3x^(2) + x + 1 at x = 2 by working througheach step of the algorithm showing the values assigned at each assignment step.b) Exactly how many multiplications and additions areused by this algorithm to evaluate a polynomial ofdegree n at x = c? (Do not count additions used toincrement the loop variable.)
- The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written recursively.Correct answer will be upvoted else downvoted. Computer science. Presently Nezzar has a beatmap of n particular focuses A1,A2,… ,An. Nezzar might want to reorder these n focuses so the subsequent beatmap is great. Officially, you are needed to find a change p1,p2,… ,pn of integers from 1 to n, to such an extent that beatmap Ap1,Ap2,… ,Apn is great. In case it is unthinkable, you ought to decide it. Input The primary line contains a solitary integer n (3≤n≤5000). Then, at that point, n lines follow, I-th of them contains two integers xi, yi (−109≤xi,yi≤109) — directions of point Ai. It is ensured that all focuses are unmistakable. Output In case there is no arrangement, print −1. In any case, print n integers, addressing a legitimate change p. In case there are numerous potential replies, you can print any.Using recursion, write a Python function def before(k,A) which takes an integer k and an array A of integers as inputs and returns a new array consisting of all the integers in A which come before the last occurrence of k in A, in the same order they are in A. For example, if A is [1,2,3,6,7,2,3,4] then before(3,A) will return [1,2,3,6,7,2]. If k does not occur in A, the function should return None.
- a) Consider a recursive function to return the Number of Binary Digits in the Binary Representation of a Positive Decimal Integer (n) using a recursive algorithm. int Process (int n) { if (n == 1) return 1; else return (Extra() + Process (n/4) + Process (n/4)); } Given that Extra(n) is a function of O(n)1) Find T(n) = number of arithmetic operations. 2) Calculate the complexity of this algorithm using Back Substitution.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.Modulo arithmeticFor two integers ‘a’ and ‘b’ and a positive integer ‘n’, let + represent the operation a + b = (a+b)mod(n), that is, the result of a + b is the remainder of the usual sum a + b after dividing by n (using a natural representation). Similarly, let * represent the operation a * b = (a*b)mod(n), that is, the result of a * b is the remainder of the product a*b after dividing by n. For example using the set S = {2, 5, 8} and n = 9, 5 + 8 = 4, since 4 is the remainder after dividing 13 (5 + 8) by 9; and 2 * 8 = 7, since 7 is the remainder after dividing 16 (2*8) by 9.a. For each set S and number n specified below, create two “operation” tables (these are called Cayley tables), one showing the results for the operation +, and one showing the results of the operation * for each pair of elements from S (see the example tables in the notes).i. S1 = {0, 1} and n = 2ii. S2 = {1, 2} and n = 3iii. S3 = {0, 2, 4, 6} and n = 8iv. S4 = {1, 3, 5, 7} and n = 8v. S5 = {1, 2, 3, 4} and n =…