Copyright © 2014 by John Wiley & Sons. All rights reserved.1 Chapter 13 - Recursion

Copyright © 2014 by John Wiley & Sons. All rights reserved. 1

Chapter 13 - Recursion


Chapter Goals

To learn to “think recursively” To be able to use recursive helper methods To understand the relationship between recursion and

iteration To understand when the use of recursion affects the

efficiency of an algorithm To analyze problems that are much easier to solve by

recursion than by iteration To process data with recursive structures using mutual

recursion


Recursion

For some problems, it’s useful to have a method call itself.

A method that does so is known as a recursive method.

A recursive method can call itself either directly or indirectly through another method.

E.g. • Fibonacci method• Factorial method• Towers of Hanoi• Merge sort• Linear search• Binary search• Quick sort


Recursion

A recursive method always contain a base case(s):• The simplest case that method can solve.

• If the method is called with a base case, it returns a result.

If the method is called with a more complex problem, it breaks problem into same but simpler sub-problems and recursively call itself on these smaller sub-problems.

The recursion step normally includes a return statement which allow the method to combine result of smaller problems to get the answer and pass result back to its caller.


Recursion

For the recursion to eventually terminate, each time the method calls itself with a simpler version of the original problem, the sequence of smaller and smaller problems must converge on a base case.

When the method recognizes the base case, it returns a result to the previous copy of the method.


Factorial Example Using Recursion: Factorial

• factorial of a positive integer n , called n!n! = n · (n – 1) · (n – 2) · … · 1

• 1! =1 • 0! =1• E.g. 5! = 5.4.3.2.1 = 120


Programming Question Write a iterative (non-recursive) method factorialIterrative in

class FactorialCalculator to calculate factorial of input parameter n.

Call this in the main method to print factorials of 0 through 50

A sample output is below:


Answer:

Try setting n=100 in main. How does the output change?

How does the output change when data type is int?

public class FactorialCalculator{ public static long factorialIterative(long n) { long factorial = n; for(int i=(int)n;i>=1;i--) factorial *= i; return factorial; } public static void main(String args[]) { for(int n=0;n<=100;n++) System.out.println(n+"!= "+factorialIterative(n)); }}


• How to write it recursively?• Recursive aspect comes from : n! = n.(n-1)!• E.g. 5!


Programming Question

Modify FactorialCalculator to add a recursive factorial method.


Answer

• Recursive method:

public static long factorial(long number){

if (number ==1 || number==0) //Base Case{

return number;}else //Recursion Step{

return number * factorial(number-1);}

}


public class FactorialCalculator{ public static long factorialIterative(int n) { int factorial = n; for(int i=(int)n;i>=1;i--) factorial *= i; return factorial; } public static long factorialRecursive(long number) { if (number ==1 || number==0) //Base Case { return number; } else //Recursion Step { return number * factorialRecursive(number-1); } } public static void main(String args[]) { for(int n=0;n<=100;n++) System.out.println(n+"!= "+factorialRecursive(n)); }}



Since factorial values tend to be large, using BigInteger rather than long as data type is more appropriate.

Modify FactorialCalculator to use BigInteger data type rather than long (modify recursive version only).


Since factorial values tend to be large, using BigInteger rather than long as data type is more appropriate:

import java.math.BigInteger;

public class FactorialCalculator{ public static BigInteger factorialRecursive(BigInteger number) { if (number.compareTo(BigInteger.ONE)==0 || number.compareTo(BigInteger.ZERO)==0) //Base Case { return number; } else //Recursion Step { return number.multiply(factorialRecursive(number.subtract(BigInteger.ONE))); } } public static void main(String args[]) { for(int n=0;n<=100;n++) System.out.println(n+"!= "+factorialRecursive(BigInteger.valueOf(n))); }}


Output:



Using recursion: Fibonacci Series

• The Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, … • Begins with 0 and 1 • Each subsequent Fibonacci number is the sum of the

previous two.

Implement the tester class FinbonacciCalculator that contains the recursive method fibonacci.• Hint: Use BigInteger as return data type • Call this method in main method to print fibonacci values of

0 to 40:


E.g. fibonacci(3)


A sample run is shown:


AnswerFibonacciCalculator.java


Java Stack and Heap

Before we look at how recursive methods are stored and processed, lets look at two important concept in java:• Java stack and heap

Stack:• A memory space reserved for your process by the OS• the stack size is limited and is fixed and it is determined in the

compiler phase based on variables declaration and other compiler options

• Mostly, the stack is used to store methods variables• Each method has its own stack (a zone in the process stack),

including main, which is also a function.• A method stack exists only during the lifetime of that method


Java Stack and Heap

Heap:• A memory space managed by the OS

• the role of this memory is to provide additional memory resources to processes that need that supplementary space at run-time (for example, you have a simple Java application that is constructing an array with values from console);

• the space needed at run-time by a process is determined by functions like new (remember, it the same function used to create objects in Java) which are used to get additional space in Heap.


Recursion and the Method-Call Stack Let’s begin by returning to the Fibonacci example

• Method calls made within fibonacci(3):

• Method calls in program execution stack:

Method stays in stack as long has not completed and returned



Write a class Sentence that define the recursive method isPalindrome() that check whether a sentence is a palindrome or not.

Palindrome: a string that is equal to itself when you reverse all characters • Go hang a salami, I'm a lasagna hog • Madam, I'm Adam


Class Skeleton is provided for you. Add a main method that create several sentences and print if they are palindromes.

public class Sentence { private String text; /** Constructs a sentence. @param aText a string containing all characters of the sentence */ public Sentence(String aText) { text = aText; } /** Tests whether this sentence is a palindrome. @return true if this sentence is a palindrome, false otherwise */ public boolean isPalindrome() { . . . }

public static void main(String args[]){

}}


Hint: Remove both the first and last characters. If they are same check if remaining string is a

palindrome.

What are the base cases/ simplest input?• Strings with a single character

• They are palindromes

• The empty string • It is a palindrome


Answer1 public class Sentence {2 3 private String text; 4 5 public Sentence(String aText) { 6 text = aText; 7 }8 9 public boolean isPalindrome() { 10 int length = text.length(); 11 12 if (length <= 1) { return true; } //BASE CASE13 14 char first = Character.toLowerCase(text.charAt(0)); 15 char last = Character.toLowerCase(text.charAt(length - 1)); 16 17 if (Character.isLetter(first) && Character.isLetter(last)) { 18 if (first == last) { 19 Sentence shorter = new Sentence(text.substring(1, length - 1)); 20 return shorter.isPalindrome(); 21 } 22 else { 23 return false;24 } 25 }26 else if (!Character.isLetter(last)) { 27 Sentence shorter = new Sentence(text.substring(0, length - 1)); 28 return shorter.isPalindrome(); 29 } 30 else { 31 Sentence shorter = new Sentence(text.substring(1)); 32 return shorter.isPalindrome(); 33 } 34 }

Sentence.java

Continued..


Answer36 public static void main(String[] args) {37 Sentence p1 = new Sentence("Madam, I'm Adam.");38 System.out.println(p1.isPalindrome());39 Sentence p2 = new Sentence("Nope!");40 System.out.println(p2.isPalindrome());41 Sentence p3 = new Sentence("dad");42 System.out.println(p3.isPalindrome());43 Sentence p4 = new Sentence("Go hang a salami, I'm a lasagna hog.");44 System.out.println(p4.isPalindrome());45 }46}


Recursion Vs Iteraion

Recursion has many negatives. • It repeatedly invokes the mechanism, and consequently the

overhead, of method calls. • This repetition can be expensive in terms of both processor

time and memory space. • Each recursive call causes another copy of the method to be

created• this set of copies can consume considerable memory space. • Since iteration occurs within a method, repeated method

calls and extra memory assignment are avoided.


Question

Answer: No, the recursive solution is about as efficient as the iterative approach. Both require n – 1 multiplications to compute n!.

You can compute the factorial function either with a loop, using the definition that n! = 1 × 2 × . . . × n, or recursively, using the definition that 0! = 1and n! = (n – 1)! × n. Is the recursive approach inefficient in this case?


Question

Answer: The recursive algorithm performs about as well as the loop. Unlike the recursive Fibonacci algorithm, this algorithm doesn’t call itself again on the same input. For example, the sum of the array 1 4 9 16 25 36 49 64 is computed as the sum of 1 4 9 16 and 25 36 49 64, then as the sums of 1 4, 9 16, 25 36, and 49 64, which can be computed directly.

To compute the sum of the values in an array, you can split the array in the middle, recursively compute the sums of the halves, and add the results. Compare the performance of this algorithm with that of a loop that adds the values.


Programming Question Permutations Using recursion, you can find all arrangements of a set

of objects.

A permutation is a rearrangement of the letters.


Programming Question E.g. Permutations

• The string "eat" has six permutations:"eat""eta""aet""ate""tea""tae”

Design a class Permutations that contains a recursive method permutations that takes a string and return a list of permutations

public static ArrayList<String> permutations(String word)

In main method call this method to print all permutations possible.


Problem: Generate all the permutations of "eat".• First generate all permutations that start with the letter 'e', then

'a' then 't'.

How do we generate the permutations that start with 'e'? • We need to know the permutations of the substring "at".• But that’s the same problem—to generate all permutations—

with a simpler input • Prepend the letter 'e' to all the permutations you found of 'at'.• Do the same for 'a' and 't'.

Provide a special case for the simplest strings. • The simplest string is the empty string, which has a single

permutation—itself.


Answer 1 import java.util.ArrayList; 2 3 /** 4 This program computes permutations of a string. 5 */ 6 public class Permutations 7 { 8 public static void main(String[] args) 9 { 10 for (String s : permutations("eat")) 11 { 12 System.out.println(s); 13 } 14 } 15 16 /** 17 Gets all permutations of a given word. 18 @param word the string to permute 19 @return a list of all permutations 20 */ 21 public static ArrayList<String> permutations(String word) 22 { 23 ArrayList<String> result = new ArrayList<String>(); 24 Continued

Permutations.java


25 // The empty string has a single permutation: itself 26 if (word.length() == 0) 27 { 28 result.add(word); 29 return result; 30 } 31 else 32 { 33 // Loop through all character positions 34 for (int i = 0; i < word.length(); i++) 35 { 36 // Form a shorter word by removing the ith character 37 String shorter = word.substring(0, i) + word.substring(i + 1); 38 39 // Generate all permutations of the simpler word 40 ArrayList<String> shorterPermutations = permutations(shorter); 41 42 // Add the removed character to the front of 43 // each permutation of the simpler word, 44 for (String s : shorterPermutations) 45 { 46 result.add(word.charAt(i) + s); 47 } 48 } 49 // Return all permutations 50 return result; 51 } 52 } 53 } 54

Continued


Program Run:

eatetaaetateteatae


References

Java How to Program, Dietel and Dietel http://www.itcsolutions.eu/2011/02/06/tutorial-java-8

-understand-stack-and-heap/

http://www.itcsolutions.eu/2011/02/06/tutorial-java-8-understand-stack-and-heap/

http://www.itcsolutions.eu/2011/02/06/tutorial-java-8-understand-stack-and-heap/

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Copyright © 2014 by John Wiley & Sons. All rights reserved.1 Chapter 13 - Recursion