Costs of Recursion

• Think about n!.
  • How could we compute this with a loop?

  • 
     fact = 1;
     for(i = 2; i <= n; i++) {
        fact *= i;
     } 
     

  • How many times will line 3 be executed?
    • n-2 times.
  • So this function is O(n)
• How about the recursive function

  • 
     if (n == 0) {
        return 1;
      } else {
        return n * Factorial(n-1);
      } 
     

  • This is a little different.
  • How many times will line 4 be called?
  • If we look at it, it will also be O(n)
• There is slightly more cost involved in calling a function
  • you will learn about this in architecture I hope.
• So the iterative version is slightly better.
  • But not enough to notice.
• How about Fib?
  • The recursive version is bad.
  • The computation of the performance is beyond the scope of this class.
    • But you can intuitively see it is O(2^n-2)
    • That is not a tight bound, but it will do.
  • Run Fib2 in the code directory.
  • Can you write a non-recursive version of this function?
  • This function turns out to be O(n).
• Compare recursive and non-recursive binary search programs.
• Recursion is a valuable problem solving tool.
  • It can be nearly as efficient as non-recursive solutions.
  • It can be more expensive as well.
• Later you will learn MANY algorithms that have recursive definitions
  • Some are very hard to make non-recursive
  • Most involve the use of a stack.