The Fibonacci numbers are defined by the recurrence if n= if n 1 rt n-1 + F-2, if n > 2 Show that Fn 22"/2, n26. (b) Assume that the cost of adding, subtracting, or multiplying two integers is O(1), inde- pendent of the size of the integers. rite pseudocode for an algorithm that computes Fn based on the recur sive definition above. Develop a recurrence for the running time of your algorithm and give an asymptotic lower bound for it. rite pseudocode for a non-recursive algorithm that asymptotically per forms fewer additions than the recursive algorithm. Discuss the running time of the new algorithm Show how to compute Fn in O(log n) time using only integer additions and multiplications (Hint: Express Fn in matrix notation and consider the matrix and its powers. Now assume that adding two m-bit integers requires ?(m) time and that multiplying two m-bit integers requires ?(m2) time. What is the running time of the three algorithms under this more reasonable cost measure for the elementary arithmetic operations? The Fibonacci numbers are defined by the recurrence if n= if n 1 rt n-1 + F-2, if n > 2 Show that Fn 22"/2, n26. (b) Assume that the cost of adding, subtracting, or multiplying two integers is O(1), inde- pendent of the size of the integers. rite pseudocode for an algorithm that computes Fn based on the recur sive definition above. Develop a recurrence for the running time of your algorithm and give an asymptotic lower bound for it. rite pseudocode for a non-recursive algorithm that asymptotically per forms fewer additions than the recursive algorithm. Discuss the running time of the new algorithm Show how to compute Fn in O(log n) time using only integer additions and multiplications (Hint: Express Fn in matrix notation and consider the matrix and its powers. Now assume that adding two m-bit integers requires ?(m) time and that multiplying two m-bit integers requires ?(m2) time. What is the running time of the three algorithms under this more reasonable cost measure for the elementary arithmetic operations