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

Copyright © 2014 by John Wiley & Sons. All rights reserved.2 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.3 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.4 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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.5 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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.6 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! = = 120

Copyright © 2014 by John Wiley & Sons. All rights reserved.7 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:

Copyright © 2014 by John Wiley & Sons. All rights reserved.8 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)); }

Copyright © 2014 by John Wiley & Sons. All rights reserved.9 How to write it recursively? Recursive aspect comes from : n! = n.(n-1)! E.g. 5!

Copyright © 2014 by John Wiley & Sons. All rights reserved.10 Programming Question  Modify FactorialCalculator to add a recursive factorial method.

Copyright © 2014 by John Wiley & Sons. All rights reserved.11 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); }

Copyright © 2014 by John Wiley & Sons. All rights reserved.12 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)); }

Copyright © 2014 by John Wiley & Sons. All rights reserved.13 Programming Question  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).

Copyright © 2014 by John Wiley & Sons. All rights reserved.14  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))); }

Copyright © 2014 by John Wiley & Sons. All rights reserved.15  Output:

Copyright © 2014 by John Wiley & Sons. All rights reserved.16 Programming Question  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:

Copyright © 2014 by John Wiley & Sons. All rights reserved.17  E.g. fibonacci(3)

Copyright © 2014 by John Wiley & Sons. All rights reserved.18  A sample run is shown:

Copyright © 2014 by John Wiley & Sons. All rights reserved.19 Answer FibonacciCalculator.java

Copyright © 2014 by John Wiley & Sons. All rights reserved.20 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.21 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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.22 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.23 Programming Question 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.24 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 aText a string containing all characters of the sentence */ public Sentence(String aText) { text = aText; } /** Tests whether this sentence is a true if this sentence is a palindrome, false otherwise */ public boolean isPalindrome() {... } public static void main(String args[]) { }

Copyright © 2014 by John Wiley & Sons. All rights reserved.25 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.26 Answer 1 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(); if (length <= 1) { return true; } //BASE CASE char first = Character.toLowerCase(text.charAt(0)); 15 char last = Character.toLowerCase(text.charAt(length - 1)); 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..

Copyright © 2014 by John Wiley & Sons. All rights reserved.27 Answer 36 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}

Copyright © 2014 by John Wiley & Sons. All rights reserved.28 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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.29 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?

Copyright © 2014 by John Wiley & Sons. All rights reserved.30 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 is computed as the sum of and , 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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.31 Programming Question  Permutations  Using recursion, you can find all arrangements of a set of objects.  A permutation is a rearrangement of the letters.

Copyright © 2014 by John Wiley & Sons. All rights reserved.32 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 permutations(String word) In main method call this method to print all permutations possible.

Copyright © 2014 by John Wiley & Sons. All rights reserved.33  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.

Copyright © 2014 by John Wiley & Sons. All rights reserved.34 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 } /** 17 Gets all permutations of a given word. word the string to permute a list of all permutations 20 */ 21 public static ArrayList permutations(String word) 22 { 23 ArrayList result = new ArrayList (); 24 Continued Permutations.java

Copyright © 2014 by John Wiley & Sons. All rights reserved // 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); // Generate all permutations of the simpler word 40 ArrayList shorterPermutations = permutations(shorter); // 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

Copyright © 2014 by John Wiley & Sons. All rights reserved.36 Program Run: eat eta aet ate tea tae

Copyright © 2014 by John Wiley & Sons. All rights reserved.37 References  Java How to Program, Dietel and Dietel  8-understand-stack-and-heap/ 8-understand-stack-and-heap/