03+-+Tail+Recursion+and+Intro+to+Testing

03+-+Tail+Recursion+and+Intro+to+Testing - Recursion...

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EECS 280 Programming and Introductory Data Structures Recursion Another kind of factorial int fact_helper(int n, int result) // REQUIRES: n >= 0 // EFFECTS: returns result * n! { if (n == 0) { return result; } else { return fact_helper(n-1, result * n); } } int factorial(int num) // REQUIRES: n >= 0 // EFFECTS: returns num! { return fact_helper(num, 1); } Re-write the recursive version to use the same amount of space as is required by the iterative version (approximately). Recursion Group Exercise: Another kind of factorial c This function is equivalent to the original factorial. c Try to come up with a proof for why. c There are two steps. First, prove the base case, and second, the inductive step. int fact_helper(int n, int result) // REQUIRES: n >= 0 // EFFECTS: returns result * n! { if (n == 0) { return result; } else { return fact_helper(n-1, result * n); } } int factorial(int num) // REQUIRES: n >= 0 // EFFECTS: returns num! { return fact_helper(num, 1); } Recursion Another kind of factorial c There is an important thing to notice about fact_helper. c For every call to fact_helper: n! * result == num! c For the first call, this is easy to see, since: n == num result == 1 int fact_helper(int n, int result) // REQUIRES: n >= 0 // EFFECTS: returns result * n! { if (n == 0) { return result; } else { return fact_helper(n-1,result*n); } } int factorial(int num) // REQUIRES: n >= 0 // EFFECTS: returns num! { return fact_helper(num, 1); } Recursion Another kind of factorial c For every call to fact_helper: n! * result == num! c For the second call: n == (num - 1) result == (1*num) == num Substituting, we get: (num-1)! * num == num! c This is true by inspection. You can continue unwinding if you like. int fact_helper(int n, int result) // REQUIRES: n >= 0 // EFFECTS: returns result * n! { if (n == 0) { return result; } else { return fact_helper(n-1,result*n); } } int factorial(int num) // REQUIRES: n >= 0 // EFFECTS: returns num! { return fact_helper(num, 1); } Recursion Another kind of factorial c For every call to fact_helper: n! * result == num! c This is called the “recursive invariant” of fact_helper and is something that is always true. c
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This note was uploaded on 10/26/2010 for the course EECS 280 taught by Professor Noble during the Fall '08 term at University of Michigan.

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03+-+Tail+Recursion+and+Intro+to+Testing - Recursion...

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