Download ppt - Dr. Wissam Fawaz

Recursion
Canonical example: factorial

Recursion
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Classical example – the factorial function
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When a method calls itself
n! = 1· 2· 3· ··· · (n-1)· n
Recursive definition
1
if n  0

f ( n)  
else
n  f (n  1)
Example I using recursion:
Sum of values from 1 to N
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Problem
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computing the sum of all the numbers between 1 and N
This problem can be recursively defined as:
N
i
 N 
i 1
N 1
i

N  N 1 
i 1
 N  N 1  N  2 
N 3
i
i 1

N 2
i
i 1
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;
}
Example I using recursion:
Sum of values from 1 to N (ctd)
result = 6
main
sum(3)
sum
result = 3
sum(2)
sum
result = 1
sum(1)
sum
Content of recursive definition
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Base case (s)
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Values of the input variables for which
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Every chain of recursive calls must
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no recursive calls are performed
 There should be at least one base case
Eventually reach a base case
Recursive calls
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Calls to the current method
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Defined in such a way that
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It makes progress towards a base case
Example II using recursion:
Factorial
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Factorial of a positive integer n, written n! is the product
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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
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iteratively (non-recursively) using a for statement as follows:
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factorial = 1;
for(int counter = number; counter >= 1; counter-- )
factorial *= counter;
Recursive declaration of the factorial method is arrived at
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by observing the following relationship:
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n! = n · (n – 1)!
Example II using recursion:
Factorial (cont’d)
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long should be used so that the program
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can calculate factorials greater than 12!
Package java.math provides classes
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BigInteger and BigDecimal explicitly for
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high precision calculations not supported by primitive types.
Refer to FactorialApp project
Example II using recursion:
Factorial (cont’d)
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BigInteger method compareTo
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compares the BigInteger that calls the method to
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the method’s BigInteger argument.
Returns
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-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.

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BigInteger constant
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ONE represents the integer value 1.
ZERO represents the integer value 0.
Example III using recursion:
Fibonacci Series
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The Fibonacci series, begins with 0 and 1 and
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has the property that each subsequent Fibonacci number
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The ratio of successive Fibonacci numbers converges to
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a constant value of 1.618…,
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is the sum of the previous two.
 0, 1, 1, 2, 3, 5, 8, 13, 21, …
called the golden ratio or the golden mean.
The Fibonacci series may be defined recursively:
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fibonacci(0) = 0
fibonacci(1) = 1
fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)
Example III using recursion:
Fibonacci Series
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Two base cases for Fibonacci method
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fibonacci(0) is defined to be 0
fibonacci(1) to be 1
Fibonacci numbers tend to become large quickly.
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We use type BigInteger as the the return type of
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BigInteger methods multiply and subtract
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The fibonacci method
implement multiplication and subtraction.
Refer to FibonacciApp project
Visualizing Recursion for Factorial
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Recursion trace
A box for each recursive call
Example recursion trace:
return 4*6 = 24
call
final answer
recursiveFactorial(4)
call
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An arrow from each caller
 to callee
return 3*2 = 6
recursiveFactorial(3)
return 2*1 = 2
call
recursiveFactorial(2)
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An arrow from each callee
 to caller showing return value
call
return 1*1 = 1
recursiveFactorial(1)
call
recursiveFactorial(0)
return 1
Example IV using recursion:
Binary Search
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A binary search
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Assumes the list of items in the search pool is sorted
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Eliminates a large part of search pool with 1 comparison
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Examines the middle element of the list
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If it matches the target, the search is over
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Otherwise, only one half of the remaining elements
 Need to be searched
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Then examines the middle element of remaining pool
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Eventually, the target is found or data is exhausted
Example IV using recursion:
Binary Search (cont’d)
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Example:
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Search a sorted array for a given value
start: index of 1st element
in search pool
middle
end: index of 1st element
in search pool
A
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BS(A, key, start, end)
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Look for key in the array A
 where elements are sorted according to ascending order
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A method that calls itself with a smaller input set
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
else
if (A[mid] > key)
return BS(A, key, start, mid-1)
else
return BS(A, key, mid+1, end)
Example V using recursion:
Towers of Hanoi
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Given
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A platform with three pegs sticking out of it
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Each peg is a stack that can accommodate n disks
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Puzzle
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Move all disks from peg a to peg c, one disk at a time
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So that we never place a larger disk on top of smaller one
Refer to TowersOfHanoiApp project
Towers of Hanoi
Original Configuration
Move 1
Move 2
Move 3
Towers of Hanoi
Move 4
Move 5
Move 6
Move 7 (done)