ΡΥTHOΝ Determine a function T(n) that relates input size n to number of runtime steps and the Big-O Time complexity for this T(n): Example only: T{n) = an^2 +k Соmplexity 3D Om) def f1(n) : print(1) print(2) T(n) = Complexity print(n-4) print(n-3)
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- Assume that each of the expressions below gives the processing time T(n) spent by an algorithm for solving a problem of size n. Select the dominant term(s) having the steepest increase in n and specify the lowest Big-Oh complexity of each algorithm. For example, the dominant term in 0.1n + 10n4 is 10n4 and it is O(n4). Expression Dominant term(s) O(. . .) 5 + 0.001n3 + 0.025n 500n + 100n1.5 + 50n log10 n 0.3n + 5n1.5 + 2.5 · n1.75 n2 log2 n + n(log2 n)2 n log3 n + n log2 n 100n + 0.01n2 0.01n + 100n2 2n + n0.5 + 0.5n1.25 0.01n log2 n + n(log2 n)2 100n log3 n + n3 + 100nBig-O notation. Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n2, f6(n) = 2n, f7(n) = n!, f8(n) = nn. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n2 < 2n < n! < nn. Prove this is true. Let f, g : N → R+, prove that Ω(f(n) + g(n)) = Ω(max{f(n), g(n)}). [Note: Proving this will help you understand that we can also leave out the insignificant parts in big-Ω notation and the result is still a lower bound, e.g., Ω(n2 + n log n + n) = Ω(n2).]Suppose that your implementation of a particular algorithm appears in C++ as for (int pass = 1; pass <= n; pass++) { for (int index = 0; index < n; index++) { for (int count = 1; count < 10; count++) { ... } // end for } // end for } // end for The previous code shows only the repetition in the algorithm, not the computations that occur within the loops. These computations, however, are independent of n. What is the Big O of the algorithm?
- 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.Let A(n) = 3A(n/2) + n2 + 3n + 4.Using the MasterTheorem, the computational complexity is _______. A.O(n2log n) B.O(n1.58) where 1.58 = log23 C.O(n2) D.O(n0.63) where 0,63 = log32The given time complexity is: m T(m-1) + ca > 1 cb = 0 expanding using substitution: m T(m-1) + ca m[(m-1) T(m-2) + ca]+ ca m(m-1) T(m-2) + mca + ca m(m-1) [(m-2) T(m-3) + ca] + mca + ca m(m-1)(m-2) T(m-3) + m(m-1)ca + mca + ca what is the time complexity? form an expression for adding all the ca
- #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.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.Information is present in the screenshot and below. Based on that need help in solving the code for this problem in python. The time complexity has to be as less as possible (nlogn or n at best, no n^2). Apply divide-and-conquer algorithm in the problem. Make sure all test cases return expected outputs. Hint: Apply bisection method/modules Output FormatOutput contains a line with two space-separated integers W_a and W_b.- W_a is the maximum matchups won by Hamiltonia- W_b is the maximum matchups won by Burrgadia. Sample Input 03 554402410 Sample Output 03 0 Sample Input 15 4833485183 Sample Output 12 2 Sample Input 27 81028121912601319851 Sample Output 27 0 The actual code """Solves a test case Parameters:a : int - number of leaders in Hamiltoniab : int - number of leaders in Burrgadias_i : array-like - rap proficiencies of Hamiltonia's leadersr_j : array-like - rap proficiencies of Burrgadia's leaders Returns:win_a : int - number of…
- Please written by computer source Let x be a binary string. The minimal description of x, written d(x), is the shortest string ⟨M, w⟩ where TM M on input w halts with x on its tape. If several such strings exist, select the lexicographically first among them. The descriptive complexity, or Kolmogorov complexity, of x, written K(x), is K(x) = |d(x)|. Show that the function K(x) is not a computable function. HINTS: If K is a computable function, there is some TM which computes it. That TM can be used to find strings of large complexity. Try to design a program which outputs “complex” strings but which contradicts their supposed complexity, and even contradicting the supposed complexity of a single string suffices.Big-O notation. (a) Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n2, f6(n) = 2n, f7(n) = n!, f8(n) = nn. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n2 < 2n < n! < nn. Prove this 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(n2), n2 = O(2n), 2n = O(n!), n! = O(nn). (b) Let f, g : N → R+, prove that O(f(n) + g(n)) = O(max{f(n), g(n)}). [Hint: The key is max{f(n), g(n)} ≤ f(n) + g(n) ≤ 2 · max{f(n), g(n)}. Note: Proving this will help you to understand why we can leave out the insignificant parts in big-O notation and only keep the dominate part, e.g., O(n2+n log n+n) = O(n2).] (c) Let f, g : N → R+, prove that Ω(f(n) + g(n)) = Ω(max{f(n), g(n)}). [Note: Proving this will help you understand…Modeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…