1. Recursion
Gerry Howser, April 23, 2015

2. Example: Linear Search of an unordered list
Iteration
Usually do a series of sequential steps on a piece of data and then repeat on a new piece of data
Often involve loops
Example: Linear Search of an unordered list

3. Iteration vs Recursion
Some problems have both iterative and recursive solutions
Some problems lend themselves to recursive solutions
Some problems do not lend themselves to recursion

4. Recursive Solutions
Only specific types of problems can have recursive solutions
The problem can be broken into sub-problems that exactly resemble the original problem
Often the problem can be split in half
Binary search
Merge sort
The solutions to the sub-problems combine to form a solution to the original problem

5. Two Quotes to Remember
“Begin at the beginning”, the King said, very gravely, “and go on till you come to the end: then stop.” Lewis Carroll
“If all you have is a hammer, everything looks like a nail.” Maslow’s Law

6. Problems with Recursive Solutions
The Tower of Hanoi
Merge Sort
Matrix Problems
Factorials
Others…..

7. Tower of Hanoi

8. Tower of Hanoi Pseudo Code
Solve(N, Src, Aux, Dst)
if N is 0
exit
else
Solve(N - 1, Src, Dst, Aux)
Move from Src to Dst
Solve(N - 1, Aux, Src, Dst)

9. Recursive Merge Sort
Merge_sort(A[ ],low, high) if low < high q = floor((low+high)/2.0) Merge_sort(A[ ], low, q) Merge_sort(A[ ], q+1, high) Merge(A[ ], low, q, high) return

10. Binary Tree of Pool Balls
Given 15 pool balls, build a tree such that:
Any parent is larger than its left child
Any parent is less than its right child
Why does this hold for 7?

11. Pool Ball Binary Tree

12. Binary Tree of Pool Balls
Problem: Write an algorithm to list the pool balls in ascending order (In-order tree walk)
Difficult to do with loops
But if we choose wisely…..
Can break the problem into identical sub-problems
RECURSION!!!!

13. Pseudo Code to List Tree In Order
list_in_order(Tree, node){ if (node.left exists) list_in_order(Tree, node.left); list(node); if(node.right exists) list_in_order(Tree, node.right); return;}

14. Listing the Tree All we need now is a way to start this at the root:
in_order(Tree,root){
list_in_order(Tree, root);
return;}

15. Factorials n!=n(n-1)(n-2)…1 (n-1)!=(n-1)(n-2)…1 n!=n * (n-1)!
All sub-problems are identical to original problem … and contribute to the solution
RECURSION!!!

16. Pseudo Code for n!
int fact(int i) { int result, n; n=i; if (n<0) result = -1; else result = 1; if(n>1) result = n * fact(n-1); return result; }

17. Recursion and n! (continued…)
Is this the only solution?
Is this the best solution?
What does “best” mean?
Is there a better solution?
Loop version

18. Loops and n!
int fact(int n) { int result; int i=n; result=1; if(i<0) result = -1; while (i>0){ result = result * I; i--; }; return result; }

19. Generic Recursion
recurse(A[ ],n) If(not base_case) recalculate n recurse(new_n) //or multiple times smash_together_answer return

20. Conclusions Many problems have recursive solutions
Must be able to break the problem into sub-problems with exactly the same solution method
Must be able to smash sub-solutions together to form the complete solution
Not all recursive solutions are best
Not all problems have recursive solutions