1. RECURSION
Annie Calpe

2. Overview Introduction Review: the Basics How it works - Examples
- Factorial
- Fibonacci Sequence
- Sierpinski Curve
Stacks
Applications
Design Considerations

3. Introduction to Recursion
Recursion can be used to manage repetition.
Recursion is a process in which a module achieves a repetition of algorithmic steps by calling itself.
Each recursive call is based on a different, generally simpler, instance.

4. Introduction to Recursion
Recursion is something of a divide and conquer, top-down approach to problem solving.
- It divides the problem into pieces or selects out one key step, postponing the rest.

5. 4 Fundamental Rules :
Base Case : Always have at least one case that can be solved without recursion.
Make Progress : Any recursive call must progress towards a base case.
Always Believe : Always assume the recursive call works.
Compound Interest Rule : Never duplicate work by solving the same instance of a problem in separate recursive calls.

6. Basic Form : void recurse () { recurse (); //Function calls itself }
int main ()
recurse (); //Sets off the recursion

7. How does it work? The module calls itself.
New variables and parameters are allocated storage on the stack.
Function code is executed with the new variables from its beginning. It does not make a new copy of the function. Only the arguments and local variables are new.
As each call returns, old local variables and parameters are removed from the stack.
Then execution resumes at the point of the recursive call inside the function.

8. Recursion Trees
A key tool for analyzing recursive algorithms is the recursion tree.
The total processing time is related to the total # of nodes
The necessary storage space is related to its height

9. To Build a Recursion Tree:
root = the initial call
Each node = a particular call
Each new call becomes a child of the node
that called it
A tree branch (solid line) = a call-return path between any 2 call instances

10. Factorial Factorial (n): IF (n = 0) RETURN 1 ELSE
RETURN n * Factorial (n-1)
Calculates n*(n-1)*(n-2)*…*(1)*(1)

11. Fibonacci Numbers F (n) = F (n-1) + F (n-2) Fibonacci (n)
IF (n <= 1)
RETURN n
ELSE
RETURN Fibonacci (n-1) +
Fibonacci (n-2)
** Inefficient use of recursion !!

12. Recursion Tree showing Fibonacci calls

13. Space Filling Curves
A continuous mapping from a lower-dimensional space into a higher-dimensional one, using fractals.
Fractals are shapes that occur inside other, similar shapes.
A useful property of a space-filling curve is that it tends to visit all the points in a region once it has entered that region.

14. The Sierpinski Curve
The “limiting curve” of an infinite sequence of curves numbered by an index n=1,2,3…
It ends up covering every point in the region.
Fills 2-D space (fills a plane using lines)

15. The Sierpinski Curve ZIG (n): if (n = 1) turn left, advance 1 else
ZAG (n/2)
ZAG (n):
if (n = 1)
turn right, advance 1
turn left
else
ZAG (n/2)
ZIG (n/2)

16. ZIG(4) – ¼ Complete ** End of first ZIG (2) call ZIG (4) ZAG ZIG

17. ZIG(4) – ½ Complete ZIG (4) ZIG (2) ZAG (2) ZIG (2) ZAG (2) ZIG (1)

18. ZIG(4) – ½ Complete ** End of first ZAG (2) call ZAG ZIG ZAG ZIG ZIG

19. ZIG(4) – The Rest? No need to go further. Why?
ANSWER: Because of Rule #3
- We’ve shown that the base case works as well as the next case.

20. Run-time Stack Use
Recursion is controlled in a computer by means of a pushdown stack.
A push = a new function call
A pop = a completed execution of
a function call
Stack overflow is possible
A move down = a push onto runtime stack
A move up = a pop & a move down = a push

21. Some Uses For Recursion
Numerical analysis
Graph theory
Symbolic manipulation
Sorting
List processing
Game playing
General heuristic problem-solving
Tree traversals

22. Why use it? PROS CONS Clearer logic Often more compact code
Often easier to modify
Allows for complete analysis of runtime performance
CONS
Overhead costs

23. Summary Recursion can be used as a very powerful programming tool
There is a tradeoff between time spent constructing and maintaining a program and the cost in time and memory of execution.

24. References The New Turing Omnibus – Dewdney
Data Structures & Problem Solving in Java – Weiss
Computing and Algorithm – Shackelford
Recursion Tutorial – National University of Ireland, Dept of I.T.