1. Intro to Computer Science CS1510 Dr. Sarah Diesburg
Wrapping up Recursion
Intro to Computer Science
CS1510
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. Recursion – one way
A class of methods exhibit recursive behavior when they can be defined by two properties:
A simple base case (or cases), and
A set of rules which reduce all other cases toward the base case. (recursive step)

4. Recursion – another way
A recursive function is:
One that calls itself
With “smaller” inputs (a recursive step)
Until those inputs reach a base case

5. 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.

6. 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).

7. How does this work in a computer?

8. The Stack
A Stack is a data structure, like a List or a Dictionary, but with a few different characteristics.
A Stack is a sequence.
A Stack only allows access to one end of its data, the top of the stack.

10. Operations pop: remove top of stack. Stack is one element smaller.
push (val): add val to the stack. Val is now the top. Stack is one element larger.
top: Reveals the top of the stack. No modification to stack.

12. Stack of Function Calls
Python maintains a stack of function calls.
If a function calls another function recursively, the new function is pushed onto the calling stack and the previous function waits.
The top is always the active function.
When a pop occurs, the function below becomes active.

13. 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)

14. 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
=15
Write a summation function
summation(lower,higher)

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

16. 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

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

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

19. 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

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

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

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

23. 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.

24. 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 I J

25. 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.

26. 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

27. 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