03/07/13

1. Recursive Definition 03/07/13 Discrete Structures (CS 173) Derek Hoiem, University of Illinois

2. Last class: induction • False induction proofs • All professors have the same color shoes. • Other good examples: http://www.math.uiuc.edu/~hildebr/347/induction3-sol.pdf overhead

3. Today’s lecture: Recursive Definitions • Be able to write a series as a recursive function • Be able to “unroll” a recursive function and find the closed form • Prove closed form of recursive function using induction • Examples, including divide and conquer algorithm

4. Defining recursive functions A recursive function has two parts: base case: recursive formula: , for n Example:  base case  recursive formula

5. Examples

6. Examples 1 1,1 Pascal’s Triangle 1 1 2,2 2,1 1 2 1 3,1 3,2 3,3 1 3 3 1

7. Examples The series: 1, 2, 2, 4, 8, 32, 64, … (each number is the product of previous two)

8. Finding closed form

9. Process for finding closed form • “Unroll” for several steps until the pattern becomes clear • Solve for • Find for base case • Substitute value for base case • Simplify • Check answer

10. Example: how many edges in a N-d hypercube?

11. Proof of closed form solution by induction Claim: For all natural ,

12. Example: divide and conquer computation • A divide and conquer algorithm recursively splits a problem of size into two smaller problems, each of size . • Suppose it costs time to make the split and when the function returns a value in constant time . • Problem: Write cost as a recursive function , solve for closed form, and prove that solution is correct.

13. Things to remember • Be able to write a series as a recursive function • Process for finding closed form • Unroll for several steps • Write in terms of • Substitute for value of that is base case • Substitute base case value(s) and solve • Closed form solutions can be proved with induction

14. Next week • Trees