1. Recursion

2. Canonical example: factorial
Recursion
When a method calls itself
Classical example – the factorial function
n! = 1· 2· 3· ··· · (n-1)· n
Recursive definition
Recursion is used for performing repetitive tasks. To illustrate recursion, let us begin with a simple example of computing the value of the factorial function. The factorial of a positive integer n, denoted n!, is defined as the product of the integers from 1 to n. If n=0, then n! is defined as 1 by convention. The factorial function can be defined in a manner that suggests a recursive formulation. Factorial(5) = 5.factorial(4) => thus we can define factorial(5) in terms of factorial(4). In general, for a positive integer n, we can define factorial(n) to be n times factorial(n-1). This leads to what we call a recursive definition.

3. Example I using recursion: Sum of values from 1 to N
Problem
computing the sum of all the numbers between 1 and N
This problem can be recursively defined as:

4. Example I using recursion: Sum of values from 1 to N (ctd)
// This method returns the sum of 1 to num
// Refer to SumApp project
public int sum (int num) {
int result;
if (num == 1)
result = 1;
else
result = num + sum (n-1);
return result; }

5. Example I using recursion: Sum of values from 1 to N (ctd)
main
sum
sum(3)
sum(1)
sum(2)
result = 1
result = 3
result = 6

6. Content of recursive definition
Base case (s)
Values of the input variables for which
no recursive calls are performed
There should be at least one base case
Every chain of recursive calls must
Eventually reach a base case
Recursive calls
Calls to the current method
Defined in such a way that
It makes progress towards a base case
The recursive definition contains one or more base cases, which are defined nonrecursively in terms of fixed quantities. In the case of factorial, n=0 is the base case. It also contains one or more recursive cases, which are defined by appealing to the definition of the function being implemented. We can illustrate the execution of a recursive function by means of a recursive trace.

7. Example II using recursion: Factorial
Factorial of a positive integer n, written n! is the product
n · (n – 1) · (n – 2) · … · 1
with 1! equal to 1 and 0! defined to be 1
The factorial of integer number can be calculated
iteratively (non-recursively) using a for statement as follows:
factorial = 1;
for(int counter = number; counter >= 1; counter-- ) factorial *= counter;
Recursive declaration of the factorial method is arrived at
by observing the following relationship:
n! = n · (n – 1)!

8. Example II using recursion: Factorial (cont’d)
long should be used so that the program
can calculate factorials greater than 12!
Package java.math provides classes
BigInteger and BigDecimal explicitly for
high precision calculations not supported by primitive types.
Refer to FactorialApp project

9. Example II using recursion: Factorial (cont’d)
BigInteger method compareTo
compares the BigInteger that calls the method to
the method’s BigInteger argument.
Returns
-1 if the BigInteger that calls the method
is less than the argument,
0 if they are equal or
1 if the BigInteger that calls the method
is greater than the argument.
BigInteger constant
ONE represents the integer value 1.
ZERO represents the integer value 0.

10. Example III using recursion: Fibonacci Series
The Fibonacci series, begins with 0 and 1 and
has the property that each subsequent Fibonacci number
is the sum of the previous two.
0, 1, 1, 2, 3, 5, 8, 13, 21, …
The ratio of successive Fibonacci numbers converges to
a constant value of 1.618…,
called the golden ratio or the golden mean.
The Fibonacci series may be defined recursively:
fibonacci(0) = 0 fibonacci(1) = 1 fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)

11. Example III using recursion: Fibonacci Series
Two base cases for Fibonacci method
fibonacci(0) is defined to be 0
fibonacci(1) to be 1
Fibonacci numbers tend to become large quickly.
We use type BigInteger as the the return type of
The fibonacci method
BigInteger methods multiply and subtract
implement multiplication and subtraction.
Refer to FibonacciApp project

12. Visualizing Recursion for Factorial
Recursion trace
A box for each recursive call
An arrow from each caller
to callee
An arrow from each callee
to caller showing return value
Example recursion trace:
recursiveFactorial ( 4 ) 3 2 1
return
call * = 6 24
final answer
Each entry of the trace corresponds to a recursive call. Each new recursive function call is indicated by an arrow to the newly called function. When the function returns, an arrow showing this return is drawn and the return value may be indicated with this arrow.

13. Example IV using recursion: Binary Search
A binary search
Assumes the list of items in the search pool is sorted
Eliminates a large part of search pool with 1 comparison
Examines the middle element of the list
If it matches the target, the search is over
Otherwise, only one half of the remaining elements
Need to be searched
Then examines the middle element of remaining pool
Eventually, the target is found or data is exhausted

14. Example IV using recursion: Binary Search (cont’d)
Search a sorted array for a given value
BS(A, key, start, end)
Look for key in the array A
where elements are sorted according to ascending order
A method that calls itself with a smaller input set
start: index of 1st element
in search pool
end: index of 1st element
in search pool
middle A

15. Binary search recursion: pseudo-code
// Refer to BinarySearchApp project
Boolean BS(A, key, start, end)
mid = (start+end)/2
if(A[mid] == key)
return true
else
if(end <= start)
return false
if (A[mid] > key)
return BS(A, key, start, mid-1)
return BS(A, key, mid+1, end)

16. Example V using recursion: Towers of Hanoi
Given
A platform with three pegs sticking out of it
Each peg is a stack that can accommodate n disks
Puzzle
Move all disks from peg a to peg c, one disk at a time
So that we never place a larger disk on top of smaller one
Refer to TowersOfHanoiApp project

17. Original Configuration
Towers of Hanoi
Original Configuration
Move 1
Move 2
Move 3

18. Towers of Hanoi
Move 4
Move 5
Move 6
Move 7 (done)