Recursive methods

Recursive methods

Recursion

• A recursive method is a method that contains a call to itself

• Often used as an alternative to iteration when iteration is awkward or “inelegant”

• Each recursive call is given a smaller portion of the original problem

• Last recursive call solves diminished problem without recursion

Example

// method writes digits of a non-negative number, stacked vertically

public void write_vertical(int n){

n = Math.abs(n);if (n < 10)

System.out.println(“” + n);else{

write_vertical(n / 10);System.out.println(“” + n % 10);

}}

Tracing recursive method write_vertical

1. write_vertical(52406); n=52406 n/10=5240

2. write_vertical(n/10) ; n=5240 n/10=524

3. write_vertical(n/10); n=524 n/10=52

4. write_vertical(n/10); n=52 n/10=5

5. write_vertical(n/10); n=5 Stop recursion!

Output:

6. System.out.println(“” + n);

5

2

4

0

67. System.out.println (“” + n%10);

8. System.out.println (“” + n%10);

9. System.out.println(n%10);

10. System.out.println (“” + n%10);

Elements of recursive method

• Base caseBase case: problem is simplified to the point where recursion becomes unnecessary: n < 10

• Variant expressionVariant expression: the part of the problem that changes, making the problem smaller: n

• Recursive callRecursive call: method calls itself: write_vertical (n / 10);

How recursion works

• Activation record: memory block that stores all the information a method needs to work:– values of local variables & parameters– where method should return to (so calling

method can resume execution)

How recursion works

• When a method call is encountered, execution of current method ceases

• Information for newly called method is stored in an activation record

• Method executes

How recursion works• If new method contains another method call,

the process just described repeats • Each recursive call generates its own

activation record– as each recursive call is encountered, previous

activation record is stored on run-time stack– when last call fails to generate a new activation

record, stacked calls are removed (in reverse of the order they were stored), and each process continues in succession

Remember, for a recursive method to be successful ...

• Must be a problem with one or more cases in which some subtasks are simpler versions of the original problem - use recursion for these

• Must also have one or more cases in which entire computation is accomplished without recursion (base case)

Another example: the powers method

• Suppose you wanted to find the value of Xn

• For most values of n, we can solve this iteratively using a simple loop:int answer = 1;for (int c = 1; c <= n; c++)

answer *= X;• We can take care of the cases of n=1 or

n=0 with simple if statements; but what about a negative value of n?

Finding a recursive solution

• We can observe that for any value of n, Xn is equal to X * X(n-1)

• Armed with this information, we can easily develop a recursive solution that covers all values of X and n

Recursive power method

double rpower (double X, int n){

if (X == 0)return 0;

else if (n == 0)return 1;

else if (n > 0)return X * rpower(X, n-1);

else // n < 0return 1 / rpower(X, -n);

}

Another example: Factorial

• The factorial of N is the product of the first N positive integers:

N * (N – 1) * (N – 2 ) * . . . * 2 * 1

• The factorial of N can be defined recursively as

1 if N = 1

factorial( N ) =

N * factorial( N-1 ) otherwise

Recursive Method

• An recursive method is a method that contains a statement (or statements) that makes a call to itself.

• Implementing the factorial of N recursively will result in the following method.

public int factorial( int N ) {

if ( N == 1 ) {

return 1;}else {

return N * factorial( N-1 );

}}

Test to stop or continue.

Test to stop or continue.

Recursive case: recursion continues.

Recursive case: recursion continues.

End case: recursion stops.

End case: recursion stops.

Another example: Anagram

• Anagram: A word or phrase formed by rearranging the letters of another word or phrase. For example, Elvis to Lives.

• We can use a recursive method to derive all anagrams of a given word.

• When we find the end case (an anagram), we will print it out.

Anagram• List all anagrams of a given word.

WordWord C A T

C T A

A T C

A C T

T C A

T A C

AnagramsAnagrams

Anagram Solution• The basic idea is to make recursive calls on a sub-word

after every rotation. Here’s how:

CC AA TT RecursionRecursion

AA TT CC

TT CC AA

RecursionRecursion

RecursionRecursion

Rotate Left

Rotate Left

C A T

C T A

A T C

A C T

T C A

T A C

4-letter word example

Anagram Algorithm

• We find the anagrams by rotating the positions of the letters.

• The trick is, when the method is called recursively, we pass a word with the first letter cut off. So the words being passed to successive calls are getting shorter and shorter.

• However, we must access all letters of the word in order to print it out.

Anagram Algorithm

• We solve this problem by passing two parameters: the prefix and the suffix of a word.

• In each successive call, the prefix increases by one letter and the suffix decreases by one letter.

• When the suffix becomes one letter only, the recursion stops.

Anagram Method

End caseEnd case

TestTest

Recursive caseRecursive case

public void anagram( String prefix, String suffix ) {String newPrefix, newSuffix;int numOfChars = suffix.length();

if (numOfChars == 1) {//End case: print out one anagramSystem.out.println( prefix + suffix );

} else {for (int i = 1; i <= numOfChars; i++ ) {

newSuffix = suffix.substring(1, numOfChars);newPrefix = prefix + suffix.charAt(0);anagram( newPrefix, newSuffix );//recursive call//rotate left to create a rearranged suffixsuffix = newSuffix + suffix.charAt(0);

}}

}

A classic: the Towers of Hanoi

• Initial situation: a stack of donut-shaped disks stacked in in order of decreasing size from bottom to top on a wooden peg

• Object: move all the disks to a second peg, one at a time– third peg available as temporary holding area– at no time may a larger disk be placed on top

of a smaller disk

Solving towers of Hanoi

• Simplest case: tower of one disk -- move disk to destination peg

• With stack of two disks:– Move top disk to third peg– Move next disk to destination– Move disk from third peg to destination

Solving towers of Hanoi

• Simplest case: tower of one disk -- move disk to destination peg

• With stack of two disks:– Move top disk to third peg– Move next disk to destination– Move disk from third peg to destination

Solving towers of Hanoi

• Simplest case: tower of one disk -- move disk to destination peg

• With stack of two disks:– Move top disk to third peg– Move next disk to destination– Move disk from third peg to destination

Solving towers of Hanoi

• Simplest case: tower of one disk -- move disk to destination peg

• With stack of two disks:– Move top disk to third peg– Move next disk to destination– Move disk from third peg to destination

Solving towers of Hanoi

• Simplest case: tower of one disk -- move disk to destination peg

• With stack of two disks:– Move top disk to third peg– Move next disk to destination– Move disk from third peg to destination

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• Stack of 3:– Move 2 disks from 1st peg to 3rd peg, using

method already described– Move 1 disk from 1st to 2nd– Move 2 disks from 3rd to 2nd

Solving towers of Hanoi

• When there are two or more disks to move, always use third peg

• With more than two disks, we can solve the problem recursively by recognizing that the pegs are interchangeable -- that is, any peg can be used as source, destination, or “spare”

Solving towers of Hanoi

• In general, for a stack of n disks:– Move n-1 disks from peg 1 to peg 3– Move 1 disk from peg 1 to peg 2– Move n-1 disks from peg 3 to peg 2

15.4 Towers of Hanoi

• The goal of the Towers of Hanoi puzzle is to move N disks from peg 1 to peg 3:

– You must move one disk at a time.– You must never place a larger disk on top of a smaller

disk.

15.4 Towers of Hanoi

• This puzzle can be solved effectively using recursion.

• The top N-1 disks must be moved to peg 2, allowing you to then move the largest disk from peg 1 to peg 3.

• You can then move the N-1 disks from peg 2 to peg 3.

15.4 Towers of Hanoipublic void towersOfHanoi (int N, int from, int to, int spare){

if (N == 1) {moveOne(from, to);

} else {towersOfHanoi(N-1, from, spare, to);moveOne (from, to);towersOfHanoi(N-1, spare, to, from);

}}private void moveOne(int from, int to) {

System.out.println(from + “--->” + to);}

When Not to Use Recursion• When recursive algorithms are designed carelessly, it can

lead to very inefficient and unacceptable solutions.• For example, consider the following:

public int fibonacci( int N ) {

if (N == 0 || N == 1) {return 1;

} else {return fibonacci(N-1) + fibonacci(N-2);

}}

Excessive Repetition• Recursive Fibonacci ends up repeating the same

computation numerous times.

Nonrecursive Fibonaccipublic int fibonacci( int N ) {

int fibN, fibN1, fibN2, cnt;

if (N == 0 || N == 1 ) {return 1;

} else {

fibN1 = fibN2 = 1;cnt = 2;while ( cnt <= N ) {

fibN = fibN1 + fibN2; //get the next fib no.fibN1 = fibN2;fibN2 = fibN;cnt ++;

}return fibN;

}}

When Not to Use Recursion

• In general, use recursion if– A recursive solution is natural and easy to

understand.– A recursive solution does not result in excessive

duplicate computation.– The equivalent iterative solution is too complex.

Preventing infinite recursion

• One-level recursion: every case is either a stopping case or makes a recursive call to a stopping case

• Since most recursive functions are, or have the potential to be, recursive beyond just one level, need more general method for determining whether or not recursion will stop

Preventing infinite recursion

• Define a variant expressionvariant expression– numeric quantity that decreases by a fixed

amount on each recursive call– in towers of Hanoi, variant expression was

height -- each recursive call used height -1 as parameter

• Base case is when variant expression is less than or equal to its threshold valuethreshold value

Proving correctness of a recursive function

• Show that there is no infinite recursion:– define the variant expression– ensure that it reaches its threshold value

• In other words, show that the function always terminates eventually