1.  2002-2010 Prentice Hall. All rights reserved. 1 CS 1120 – Computer Science II Recursion (Ch.15) Many slides modified by Prof. L. Lilien (even many without an explicit message). Slides added by L.Lilien are © 2006-2009 Leszek T. Lilien. Permision to use for non-commercial purposes slides added by L.Lilien’s will be gladly granted upon a written (e.g., emailed) request.

2.  2002-2010 Prentice Hall. All rights reserved. 2 Recursion Recursive methods - call themselves: –Directly –Indirectly Call others methods which call it Continually breaks problem down to simpler forms –Each method call divides the problem into two pieces: A piece that the method knows how to do A recursive call (a.k.a. recursion step) that solves a smaller problem In other words, each recursive method: –Waits till recursive call made by it finishes –Then finishes itself The method must converge in order to end recursion –Must include a “terminating condition” (see below)

3.  2002-2010 Prentice Hall. All rights reserved. 3 Recursion Example 1: Factorials Fig. 7.13 | Recursive evaluation of 5! Recursive formula for factorial calculation: n! = n. (n – 1)! Termination conditions: 1! = 1 0! = 1

4.  2002-2010 Prentice Hall. All rights reserved. 4 import javax.swing.JOptionPane; /** This program demonstrates the recursive factorial method. */ public class FactorialDemo { public static void main(String[] args) { String input; // To hold user input int number; // To hold a number // Get a number from the user. input = JOptionPane.showInputDialog("Enter a " + "nonnegative integer:"); number = Integer.parseInt(input); // Display the factorial of the number. JOptionPane.showMessageDialog(null, number + "! is " + factorial(number)); System.exit(0); } /** The factorial method uses recursion to calculate the factorial of its argument, which is assumed to be a nonnegative number. @param n The number to use in the calculation. @return The factorial of n. */ private static int factorial(int n) { if (n == 0) return 1; // Base case else return n * factorial(n - 1); } } // end of class FactorialDemo Sample I/O: The recursion ends when the number is 0 Main calls the factorial method to get the result Factorial calls itself recursively here Must be static since it is called by a static method ( main ). Recall that a static method can only use/call static fields/methods.

5.  2002-2010 Prentice Hall. All rights reserved. 5 Run FactorialDemo and watch Call Stack in Debugging window 1) Open Output window, Debugging>>Variables window 2) Set up breakpoint in line 37 (left click on left gray margin) 3) Select Debug>>Debug Main Project 4) Enter input (e.g., 4) – as requested 5) Open Hidden Source Calls folder (bottom of the Debugging window) 6) Look at value of n in Variable window 7) Select Debug>>Continue (or hit F5) 8-N) Repeat 5-7 till execution finishes Final notes for FactorialDemo I bet that you could write a program calculating n! for n = 0, 1, …, 10 without using recursion that is faster and requires less memory (a more efficient program) We use recursion here (in this case less efficient) for educational purposes So that you can learn how to use recursion in these cases when using recursion is the best way to solve a problem © Leszek T. Lilien, 2009-2012

6.  2002-2010 Prentice Hall. All rights reserved. 6 ** READ LATER** Recursion (Cont.) Common Programming Error Infinite recursion (error!) occurs when the programmer either: 1)omits the base case (terminating condition), or 2)writes the recursion step incorrectly so that it does not converge on the base case Infinite recursion will eventually exhaust available memory. This error is analogous to the problem of an infinite loop in an iterative (nonrecursive) solution.

7.  2002-2010 Prentice Hall. All rights reserved. 7 Recursion Example 2: The Fibonacci Numbers Fibonacci Numbers (a.k.a. Fibonacci Sequence) –Recursion used to calculate F(n): F(0) = 0 F(1) = 1 F(n) = F(n - 1) + F(n - 2) Example – –Set of recursive calls to method Fibonacci [from Fig. 6.17 (ed.1)] return 1return 0 F( 1 )F( 0 )return 1 F( 3 ) F( 2 )F( 1 ) + return + Two terminating conditions. }

8.  2002-2010 Prentice Hall. All rights reserved. import javax.swing.JOptionPane; /** This program demonstrates the recursive fib method. */ public class FibNumbers { public static void main(String[] args) { String input; // To hold user input int number; // To hold a number // Get a number from the user. input = JOptionPane.showInputDialog("Enter a " + "nonnegative integer:"); number = Integer.parseInt(input); // Display the Fibonacci number for the input. JOptionPane.showMessageDialog(null, "Fibonacci(" + number + ") is " + fib(number)); System.exit(0); } /** The fib method calculates the nth number in the Fibonacci series. @param n The nth number to calculate. @return The nth number. */ public static int fib(int n) { if (n == 0) return 0; else if (n == 1) return 1; else return fib(n - 1) + fib(n - 2); } } // end of class FibNumbers 8 Main calls the recursive fib method to get result The recursion ends (“return”) when the number is 0 or 1 Fibonacci calls itself twice, to get the result as the sum of the two previous Fibonacci numbers Must be static since it is called by a static method ( main ).

9.  2002-2010 Prentice Hall. All rights reserved. 9 Output example 1: Output example 2: © Leszek T. Lilien, 20012

10.  2002-2010 Prentice Hall. All rights reserved. 10 Run FibNumbers and watch Call Stack in Debugging window 1) Open Output window, Debugging>>Variables window 2) Set up breakpoint in line 33 (left click on left gray margin) 3) Select Debug>>Debug Main Project 4) Enter input (e.g., 6) – as requested 5) Open Hidden Source Calls folder (bottom of the Debugging window) 6) Look at value of n in Variable window 7) Select Debug>>Continue (or hit F5) 8-N) Repeat 5-7 till execution finishes Final notes for FibNumbers (analogous as for n!) I bet that you could write a program calculating Fibonacci(n) without using recursion that is faster and requires less memory (a more efficient program) We use recursion here (in this case less efficient) for educational purposes So that you can learn how to use recursion in these cases when using recursion is the best way to solve a problem © Leszek T. Lilien, 2009

11.  2002-2010 Prentice Hall. All rights reserved. 11 Recursion vs. Iteration Iteration –Uses repetition structures while, do/while, for, foreach –Continues until counter fails repetition condition Recursion –Uses selection structures if, if/else, switch –Repetition through method calls –Continues until a termination case reached (e.g., fib(0) or fib(1 ) for Fibonacci numbers progam) –Creates duplicates of the variables Can consume a lot of memory and processor speed

12.  2002-2010 Prentice Hall. All rights reserved. 12 Recursion Example 3: Towers of Hanoi Background: –In a temple in the Far East, priests are attempting to move a stack of 64 disks from one peg to another –The legend: The world will end when the priests complete their task –The initial stack of 64 disks is threaded onto Peg 1 and arranged from bottom to top by decreasing size –The priests are attempting to move the stack from this peg to Peg 3 under the constraints that: Exactly one disk is moved at a time and At no time may a larger disk be placed above a smaller disk –Peg 2 is available for temporarily holding disks

13.  2002-2010 Prentice Hall. All rights reserved. 13 Towers of Hanoi – Cont. Write an application to solve the Towers of Hanoi problem. Allow the user to enter the number of disks. Use a recursive Tower method with four parameters: a) the number of disks to be moved, b) the id of the initial peg (on which these disks are initially threaded), c) the id of the final peg (to which this stack of disks is to be moved), d) the id of the temporary peg (to be used as a temporary holding area). The application should display the precise instructions it will take to move the disks from the starting peg to the destination peg. — Example: To move a stack of 3 disks from Peg 1 to Peg 3, the application should display the following series of moves: 1 --> 3 (This notation means “Move the top disk from Peg 1 to Peg 3.”) 1--> 2 3 --> 2 1 --> 3 2 --> 1 2 --> 3 1 --> 3

14.  2002-2010 Prentice Hall. All rights reserved. 14 Towers of Hanoi – Cont. Developing an algorithm that displays the precise sequence of peg-to-peg disk transfers If we were to approach this problem with conventional methods (e.g. using iteration), we would rapidly find ourselves hopelessly knotted up in managing the disks If we attack the problem with recursion in mind, it immediately becomes quite easy to solve

15.  2002-2010 Prentice Hall. All rights reserved. 15 Towers of Hanoi – Cont. Design -- Approach: 1) Moving n disks from Peg 1 to Peg 3, using Peg 2 as a temporary peg can be viewed recursively in terms of moving only n – 1 disks: a) Move n – 1 disks from Peg 1 to Peg 2, using Peg 3 as a temporary peg b) Move the last disk (the largest) from Peg 1 to Peg 3 c) Move the n – 1 disks from Peg 2 to Peg 3, using Peg 1 as a temporary peg 2) Terminating condition (base case ) : The process ends when the last task requires moving n = 1 disks This last task is accomplished by simply moving the disk (from the initial to the final peg) without the need for a temporary holding area

16.  2002-2010 Prentice Hall. All rights reserved. 16 Method main calls the recursive method Hanoi to get result import java.util.Scanner; /** Application solves the Towers of Hanoi problem using recursion. */ public class TowerOfHanoi { static public void main(String[] args) { // Create a Scanner object for keyboard input. Scanner keyboard = new Scanner(System.in); // Get the number of disks from the user. System.out.print("Enter the number of disks for Towers of Hanoi: "); int nrDisks = keyboard.nextInt(); // Call the recursive method ToH (Towers of Hanoi) ToH(nrDisks, 1, 3, 2); } // end method main /** The ToH method displays all disk moves. @param num The number of discs to move. @param fromPeg The peg to move from. @param toPeg The peg to move to. @param tempPeg The temporary peg. */ private static void ToH(int num, int fromPeg, int toPeg, int tempPeg) { if (num > 0) { ToH(num - 1, fromPeg, tempPeg, toPeg); System.out.println("Move a disc from peg " + fromPeg + " to peg " + toPeg); ToH(num - 1, tempPeg, toPeg, fromPeg); } } // end method ToH } // end class TowerOfHanoi Towers of Hanoi Code main calls the recursive ToH method to get the sequence of disk moves. ToH calls itself twice, to get the result s (see pseudocode)

17.  2002-2010 Prentice Hall. All rights reserved. 17 Peg A Peg B Peg C Towers of Hanoi – 2 Sample Outputs

18.  2002-2010 Prentice Hall. All rights reserved. 18 Towers of Hanoi – Cont. Note: Don’t try to run the following program for values larger than 12 – you will use a lot of time and display (or printout) space. We’ll learn later how complexity theory explains this

19.  2002-2010 Prentice Hall. All rights reserved. 19 Final notes for Towers of Hanoi I bet that you could not write a program solving Towers of Hanoi without using recursion (even a less efficient one) We use recursion here not just for educational purposes This is an example of a problem when using recursion is the best way to solve a problem © Leszek T. Lilien, 2009

20.  2002-2010 Prentice Hall. All rights reserved. 20 The End