recursion. canonical example: factorial recursion when a method calls itself classical example –...

Recursion

Canonical example: factorial

Recursion When a method calls itself

Classical example – the factorial function n! = 1· 2· 3· ··· · (n-1)· n

Recursive definition

elsenfn

nnf

)1(

0 if1)(

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:

3

1

2

1

1

11

21

1

N

i

N

i

N

i

N

i

iNNN

iNNiNi

Example I using recursion: Sum of values from 1 to N (ctd)

// This method returns the sum of 1 to num// Refer to SumApp projectpublic 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)

main

sum

sum

sum

sum(3)

sum(1)

sum(2)

result = 1

result = 3

result = 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

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)!

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

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.

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) = 0fibonacci(1) = 1fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)

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

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)





return 1

call

call

call

call

return 1*1 = 1

return 2*1 = 2

return 3*2 = 6

return 4*6 = 24 final answercall

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

Example IV using recursion: Binary Search (cont’d) Example:

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 poolmiddle

A

Binary search recursion: pseudo-code// Refer to BinarySearchApp project

Boolean BS(A, key, start, end)mid = (start+end)/2if(A[mid] == key)

return trueelse

if(end <= start)return false

elseif (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 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

Towers of Hanoi

Original Configuration Move 1

Move 3Move 2

Towers of Hanoi

Move 4 Move 5

Move 6 Move 7 (done)

recursion. canonical example: factorial recursion when a method calls itself classical example –...

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