Vhich of the following are true, explain why: 1. log(n) = O(n) 2. n= 0(log n) 3. log? (n) = O(n) 4. n= 0(log?n) 5. log(n) = 2(n)

For each of the following function, indicate the class Q(n) the function belongs (use the simplest g(n) possible in your answer.         a) (n + 1)10                                    b) 2nlog2(n+2)2 + (n+2)2log2(n/2)                             c) 2n+1+ 3n-1                                 d)   (n 2 / 2) - 4n

DIS-algo (n): =========== r = 0 in n while i 1 do for j = 1 to i do r = r + 1 i = 1/2 return r In the above, i/2 is integer division, i.e., it returns an integer, truncating any decimal part of the result of the division- so, as one concrete example (not necessarily one that comes up in this exercise!), 5/2 would return 2, not 2.5. (a) What is the final value of r returned by this function? Express it as a function of n. (b) Express the running time (as a function of n) of DIS-algo using e-notation. (c) Express the space complexity (as a function of n) of DIS-algo using -notation. As usual, be sure to give concise but convincing explanations for your answers. NOTE: Here are some hints, which might be especially helpful for part la of this exercise: • You are welcome to assume that the input value n is a power of 2, if it simplifies your analysis. •Using summation notation may help. There's a common "trick" in algorithm analysis involving summation- for some exponent x, = (). Here's a…

Which of the following is the recursive definition for the function f(n)=2n with initial condition f(1) = 2?     f(n) = 2n + f(n-1)     f(n) = 2n - f(n-1)     f(n) = 2n * f(n-1)     f(n) = 2f(n − 1) Which of the following is the recursive definition for the function f(n)=5n+2 with initial condition f(1)=7?     f(n) = 5n - f(n-1)     f(n) = 5f(n-1) + 2     f(n) = 5n + f(n-1)     f(n) = f(n-1) + 5

In questions 4-10 estimate the Big O value by analyzing the code. Note the algorithms are written in English. Hint: you are interested in the number of operations for each algorithm.

Please written by computer source

Which among the following is the recursive definition of Factorial , i.e., n! ? f(0) = 1, f(n) = (n)f(n – 1), where n E Z andn 2 1. %D f(0) = 1, f (n) = (n + 1)f(n – 1), where n E Z and n > 1. f(0) = 0, f(n) = (n)f(n – 1), where n E Z and n > 1. %3D f (0) = 0, f(n) = (n + 1)f(n – 1), where n E Z and n 2 1.

1. Consider the following recursive function: def foo(n):   if (n == 0):      return 0   return n + foo(n - 1)a. What is foo(5)   b. What is foo(10)   c.  Suppose that n + foo(n - 1) is changed to n * foo(n - 1). What is foo(5) now?   d. What happens if foo(-1) is called?

hi,it is wrong agai n      fib(n)   2           1   2           1   2           1   2           1   2           1Number of recursive calls: 5   but have to:

Using recursion, write a Java program that takes an input ‘n’ (a number) from a user to calculate and print out the Fibonacci using the following modified definition:   F(N) = 1                                                    if n = 1 or n = 2         = F((n+1)/2)2 + F((n-1/2)2           if n is odd         = F(n/2 + 1)2 – F(n/2 – 1)2         if n is even                                     Your solution must implement recursion to receive points for this question.