1. Fibonacci numbers Fibonacci numbers:Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... where each number is the sum of the preceding two. Recursive definition: –F(0) = 0; –F(1) = 1; –F(number) = F(number-1)+ F(number- 2);

3. Redundant Calculations I To compute fib(n), we recursively compute fib(n-1). When the recursive call return, we compute fib(n-2) using another recursive call –We have already computed fib(n-2) in the process of computing fib(n-1) –We make two calls to fib(n-2)

4. Redundant Calculations II Making two method calls would double the running time Compounding effect: each recursive call does more and more redundant work –Each call to fib(n-1) and each call to fib(n-2) makes a call to fib(n-3); there are 3 calls to fib(n-3) –Each call to fib(n-2) or fib(n-3) results in a call to fib(n-4), so 5 calls to fib(n-4)

5. Redundant Calculations III C(n): number of calls to fib method C(0)=C(1)=1; For n>=2, we call fib(n) and plus all the calls needed to evaluate fib(n-1) and fib(n- 2) recursively and independently; so C(n)=c(n-1)+c(n-2)+1 The recursive routine fib is exponential

6. Analyzing the Binary Recursion Fibonacci Algorithm Let n k denote number of recursive calls made by BinaryFib(k). Then –n 0 = 1 –n 1 = 1 –n 2 = n 1 + n 0 + 1 = 1 + 1 + 1 = 3 –n 3 = n 2 + n 1 + 1 = 3 + 1 + 1 = 5 –n 4 = n 3 + n 2 + 1 = 5 + 3 + 1 = 9 –n 5 = n 4 + n 3 + 1 = 9 + 5 + 1 = 15 –n 6 = n 5 + n 4 + 1 = 15 + 9 + 1 = 25 –n 7 = n 6 + n 5 + 1 = 25 + 15 + 1 = 41 –n 8 = n 7 + n 6 + 1 = 41 + 25 + 1 = 67. Note that the value at least doubles for every other value of n k. That is, n k > 2 k/2. It is exponential!

8. n n-2 n-3 n-2 n-1 n-4 n-3 1 0 … Height=n, #nodes=2 n, complexity=O(2n)

9. A Better Fibonacci Algorithm Use linear recursion instead: Algorithm LinearFibonacci(k): Input: A nonnegative integer k Output: Pair of Fibonacci numbers (F k, F k-1 ) if k = 1 then return (k, 0) else (i, j) := LinearFibonacci(k - 1) return (i +j, i) Linear recursion: a method makes at most one recursive call each time it is invoked. Runs in O(k) time.

10. Dynamic Programming – Example Dynamic programming version of fibonacci(n) –If n is 0 or 1, return 1 –Else solve fibonacci(n-1) and fibonacci(n-2) Look up value if previously computed Else recursively compute –Find their sum and store –Return result Dynamic programming algorithm  O(n) time –Since solving fibonacci(n-2) is just looking up value