 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 © 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., ed) request.

 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)

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

 Prentice Hall. All rights reserved. 4 import javax.swing.JOptionPane; /** This program demonstrates the recursive factorial method. */ public class Factorial { 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 n The number to use in the 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.

 Prentice Hall. All rights reserved. 5 Run Factorial in Eclipse & watch Call Stack in the Debug window 1) Open Debug perspective (using icon on the perspective bar) 2) Set up breakpoints: a)In Line 22: “ JOptionPane.showMessageDialog(…); “ b)In Line 39: “ return n * factorial(n - 1); “ (to do this, right click just to the left of the line number, then select Toggle Breakpoint) 3) Select Run>>Debug 4) Enter input (e.g., 4) into the pop-up window and click “OK” 5) Program progresses to the first breakpoint 6) Look at value of input and number in the Variables window 7) Select the Resume button in the main toolbar. 8) Program progresses to the next breakpoint 9) Look at value of n in the Variables window 10-N) Repeat 7-9 till execution finishes © Leszek T. Lilien

 Prentice Hall. All rights reserved. 6 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

 Prentice Hall. All rights reserved. 7 ** 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.

 Prentice Hall. All rights reserved. 8 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 (ed.1)] return 1return 0 F( 1 )F( 0 )return 1 F( 3 ) F( 2 )F( 1 ) + return + Two terminating conditions. }

 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 n The nth number to 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 9 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 ). Sample I/O:

 Prentice Hall. All rights reserved. 10 Output example 1 (same as on the previous page): Output example 2: © Leszek T. Lilien

 Prentice Hall. All rights reserved RUN ON YOUR OWN ++ Run FibNumbers in Eclipse & watch Call Stack in the Debug window 1) Open Debug perspective (using icon on the perspective bar) 2) Set up breakpoint in Line 33: “ return fib(n - 1) + fib(n - 2 );“ (to do this, right click just to the left of the line number, then select Toggle Breakpoint) 3) Select Run>>Debug 4) Enter input (e.g., 6) into the pop-up window and click “OK” 5) Program progresses to the first breakpoint 6) Look at value of n in the Variables window 7) Select the Resume button in the main toolbar. 8-N) Repeat 5-7 till execution finishes © Leszek T. Lilien

 Prentice Hall. All rights reserved. 12 Final notes for FibNumbers (analogous as for Factorial) 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

 Prentice Hall. All rights reserved. 13 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

 Prentice Hall. All rights reserved. 14 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 Peg 1 to Peg 3 under the constraints that: Exactly one disk is moved at a time and A larger disk may never be placed above a smaller disk –Peg 2 is available for temporarily holding disks

 Prentice Hall. All rights reserved. 15 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--> > > > > > 3

 Prentice Hall. All rights reserved. 16 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

 Prentice Hall. All rights reserved. 17 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

 Prentice Hall. All rights reserved. 18 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 num The number of discs to fromPeg The peg to move toPeg The peg to move 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) RECALL (the previous page): Moving n disks from Peg 1 to Peg 3, using Peg 2 as a temporary peg : 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

 Prentice Hall. All rights reserved. 19 Peg A Peg B Peg C Towers of Hanoi – 2 Sample Outputs © Leszek T. Lilien

 Prentice Hall. All rights reserved. 20 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 © Leszek T. Lilien

 Prentice Hall. All rights reserved. 21 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

 Prentice Hall. All rights reserved. 22 The End