1. Recursive Thinking

2. The Factorial Function
n!= n(n-1)(n-2)…. 3 2 1
The standard Factorial Function written without recursion is provided below:
// The Non-Recursive Factorial Function
public static int simpleFactorial( int n ) {
if (n < 0) return 0; // Make sure n >= 0
int product = 1; // “Base Case” Product is one.
for (int k = 1; k < n; k ++)
product *= k;
return product; }

3. The Factorial Function
n!= n(n-1)(n-2)…. 3 2 1
n!= n  (n-1)!
Can break a difficult problem ( n! ) into a smaller, more manageable problem of a similar structure ( (n-1)! is now easier than calculating n!)
(n-1)!

4. What Is Recursion? It is a problem-solving process
Breaks a problem into identical but smaller problems
Eventually you reach a smallest problem
Answer is obvious or trivial
Using that solution enables you to solve the previous problems
Eventually the original problem is solved

5. Recursive Factorial Function
// A Recursive Factorial Function
public static int factorial( int n ) {
if (n < 0) return 0; // Make sure n >= 0
if (n == 0 || n == 1) return 1; // Base Cases
return ( n * factorial (n-1) ); // Recursive Call }
Conclusion: recursion splits a problem Into one or more simpler versions of itself

6. Is it possible to solve any problem recursively?

7. Requirements for Recursive Solution
At least one “small” case that you can solve directly (base case)
A way of breaking a larger problem down into:
One or more smaller sub-problems
Each of the same kind as the original
A way of combining sub-problem results into an overall solution to the larger problem

8. Writing a Recursive Method
Method definition must provide parameter
Leads to different cases
Typically includes an if or a switch statement
One or more of these cases should provide a non recursive solution
The base or stopping case
One or more cases includes recursive call
Takes a step towards the base case

9. Recursive Design Example
Write a recursive algorithm for finding length of a string
‘H’ ‘e’ ‘l’ ‘l’ ‘o’ ‘\0’

10. Recursive algorithm for finding the length of a string
if string is empty (no characters)
return  base case
else  recursive case
compute length of string without first character
return 1 + that length

11. Recursive algorithm for finding length of a string
public static int length (String str) {
if (str == null ||
str.equals(“”))
return 0;
else
return length(str.substring(1)) + 1; }

12. Recursive algorithm for finding length of a string
Overall result
length(“ace”) 3
return 1 + length(“ce”) 2
return 1 + length(“e”) 1
return 1 + length(“”)

13. Simple recursive example
void test(int i) { if (i > 1) test(i / 2); } System.out.print("*"); How many asterisks are printed by the method call test(5)?

14. Carrano, Data Structures and Abstractions with Java, Second Edition
Towers of Hanoi
The initial configuration of the Towers of Hanoi for three disks
Carrano, Data Structures and Abstractions with Java, Second Edition

15. Carrano, Data Structures and Abstractions with Java, Second Edition
Towers of Hanoi
Rules for the Towers of Hanoi game
Move one disk at a time. Each disk you move must be a topmost disk.
No disk may rest on top of a disk smaller than itself.
You can store disks on the second pole temporarily, as long as you observe the previous two rules.
Carrano, Data Structures and Abstractions with Java, Second Edition

16. Carrano, Data Structures and Abstractions with Java, Second Edition
Towers of Hanoi
The sequence of moves for solving the Towers of Hanoi problem with three disks.
Carrano, Data Structures and Abstractions with Java, Second Edition

17. Carrano, Data Structures and Abstractions with Java, Second Edition
Towers of Hanoi
The smaller problems in a recursive solution for four disks
Carrano, Data Structures and Abstractions with Java, Second Edition

18. Towers of Hanoi
Algorithm solveTowers (numberOfDisks, startPole, tempPole, endPole)
if (numberOfDisks == 1)
Move disk from startPole to endPole
else {
solveTowers (numberOfDisks - 1, startPole, endPole, tempPole)
solveTowers (numberOfDisks - 1, tempPole, startPole, endPole) }
The smaller problems in a recursive solution for four disks
Carrano, Data Structures and Abstractions with Java, Second Edition

19. Debugging a recursive method
If a recursive method does not work, check the following:
Does the method have at least one input value?
Does the method contain a statement that tests an input value and leads to different cases?
Did you consider all possible cases?
Does at least one of these cases cause at least one recursive call?
Does these recursive calls get closer to the base case?
Did you consider a base case?
Carrano, Data Structures and Abstractions with Java, Second Edition

20. How to prove the correctness of a recursive algorithm?

21. Correctness of a recursive algorithm
From Algorithm theory: an algorithm is correct if it satisfies the following TWO conditions:
Produces correct results for all valid inputs
Always terminates

22. Proof by Induction Prove the theorem for the base case(s): n=0
Show that:
If the theorem is assumed true for n,
Then it must be true for n+1
Result: Theorem true for all n ≥ 0.

23. Correctness of a recursive algorithm
Recursive proof is similar to induction:
Show base case recognized and solved correctly
Show that
If all smaller problems are solved correctly,
Then original problem is also solved correctly
Show that each recursive case makes progress towards the base case  termination properly

24. Pros and Cons of recursive methods
Recursive code is often simpler than iterative (easier to write, read, and debug)
Slower than iterative