Wrapping up Recursion

1. Wrapping up Recursion Intro to Computer Science CS1510, Section 2 Dr. Sarah Diesburg

2. Recursion • Recursive functions are functions that call themselves • We can create recursive functions by breaking them up into two smaller parts

3. 1) Call same function with something “smaller” • Recursion is a natural outcome of a divide and conquer approach to problem solving. • A recursive function defines how to break a problem down (divide) and how to reassemble (conquer) the sub-solutions into an overall solution.

4. 2) Base Case • Recursion is a process not unlike loop iteration. • You must define how long (how many iterations) recursion will proceed until it stops. • The base case defines this limit. • Without the base case, recursion will continue infinitely (just like an infinite loop).

5. Some examples of things that can be done recursively • Summation of numbers – sum(lower,upper) • Exponents - power(base,exp) • Reverse a string – reverse(string) • Merge Sort – mergeSort(lyst)

6. Summation • Usually with a summation, we have a lower bound and upper bound • We want to sum from the lower bound to the upper bound • summation(1,5) would yield • 1+2+3+4+5=15 • Write a summation function • summation(lower,higher)

7. Summation def summation(lower,higher): if lower==higher: return lower else: return lower + summation(lower+1,higher)

8. Power • Write a function that takes a base and a power and returns the result • For example, we know that 24 is 16 • 2 is the base • 4 is the power • We break it up as 2 x 2 x 2 x 2 = 16 • Same as 2 x 23 = 16

9. Power def power(base,exp): if (exp==0): return 1 else: return base*power(base,exp-1)

10. Reverse a String def reverse(string): if (len(string))==0: return "" else: return string[-1] + reverse( string[0:-1] )

11. Merge Sort • Let’s say I have 16 programming assignments, and I need to sort them in alphabetical order • I’m lazy, so I hand off half of the assignments to one student to sort, and the other half to another student to sort

12. What if everyone hands off the work? A 16 B C

13. What if everyone hands off the work? A 16 B C 8 8 D E F G

14. What if everyone hands off the work? A 16 B C 8 8 D E F G 4 4 4 4 H 2 I 1

15. Merge Sort • At some point, the last students will only have 1 paper. • They can then hand those single papers back to the student that delegated the work in the first place. • The delegating student can then sort the two piles (of 1) papers by performing a merge.

16. Merge Sort • Students I and J hand each of their sorted, 1 item stacks to H. • H performs a merge sort on the two stacks of items H 1 1 J I

17. What is a merge? • Start with two sorted piles (pile A and pile B) • Take the smallest item off the top of pile A or B and place it in the finished pile. • Repeat the previous step until no more items in either pile. • Resulting finished pile will be sorted.

18. Finishing the merge sort • All students hand off sorted piles to delegating students, each of which performs a merge sort • Finally the student that started everything (A) will merge sort two large piles of items and created the final sorted output pile

19. Pseudocode for Merge Sort • Has to be some sort of recursive algorithm • What is the base case? • If you have pile of size 1, it is sorted and you are done. • What is the recursive case? • This is trickier • We can merge two sorted piles, but we want the two piles to be sorted first • Each pile is of size n/2, where n is the total number of items we have