Mastering Proof Techniques & Recursion: Fundamentals & Examples

1. Proof Techniques and Recursion

2. Proof Techniques • Proof by induction • Step 1: Prove the base case • Step 2: Inductive hypothesis, assume theorem is true for all cases up to limit k • Step 3: Show theorem is true for next case (k+1) • Proof by counterexample • Prove a statement false by providing a case where it is not correct • Proof by contradiction • Assume theorem is false and show that the assumption implies a known property is false

3. Recursion • “A function that is defined in terms of itself is called recursive.” • Example: n! • 1*2*3*…*(n-1)*n • n*factorial of (n-1)

4. Rules of Recursion • Base cases. You must always have some bases cases, which can be solved without recursion. • Making progress. For the cases that are to be solved recursively, the recursive call must always be to a case that makes progress toward a base case. • Design rule. Assume that all the recursive calls work. • Compound interest rule. Never duplicate work by solving the same instance of a problem in separate recursive calls.

5. Function: factorial int factorial(int n) {if(n == 1) return 1; else return (n*factorial(n-1)); }

6. Iterative Factorial int factorial(int n) {int i; int product = 1; for(i = n; i > 1; i--) { product = product * i; } return product; }

7. Linear Recursion • At most 1 recursive call at each iteration • Example: Factorial • General algorithm • Test for base cases • Recurse • Make 1 recursive call • Should make progress toward the base case • Tail recursion • Recursive call is last operation • Can be easily converted (should be!) to iterative

8. Higher-Order Recursion • Algorithm makes more than one recursive call • Binary recursion • Halve the problem and make two recursive calls • Example? • Multiple recursion • Algorithm makes many recursive calls • Example?

9. Example • Design a recursive program to produce the following output: 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0

10. Analyzing Recursive Functions • Factorial • Similar to analyzing a loop • Previous example • Version 3 of Maximum Subsequence Sum problem (pg. 53)

11. Algorithm • find max sum of first half • find max sum of second half • starting from middle, iterate through list toward beginning and stop when max found • starting from middle, iterate through list toward beginning and stop when max found • return max of first half, second half, and sum of overlap

12. Recurrence • T(1) = 1 • T(N) = 2T(N/2) + O(N) • T(1) = 1, T(2) = 4 = 2*2, T(4) = 12 = 4*3, T(8) = 32 = 8*4, T(16) = 80 = 16*5 • T(N) = N*(k+1) -- if N=2k • T(N) = 2k*(k+1) • T(N) = k2k + 2k • T(N) = logN(N) + N = NlogN + N = O(NlogN)