1 1992-2007 Pearson Education, Inc. All rights rese 1 5 Recursion
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We must learn to explore all the options and possibilities that confront us in a complex and rapidly changing world. —James William Fulbright
O! thou hast damnable iteration, and art indeed able to corrupt a saint. —William Shakespeare
It's a poor sort of memory that only works backwards.—Lewis Carroll, Alice in Wonderland
Life can only be understood backwards; but it must be lived forwards. —Soren Kierkegaard
Push on—keep moving. —Thomas Morton
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OBJECTIVES
In this chapter you will learn: The concept of recursion. How to write and use recursive methods. How to determine the base case and recursion step in a
recursive algorithm. How recursive method calls are handled by the system. The differences between recursion and iteration, and
when it is appropriate to use each. What geometric shapes called fractals are and how to
draw them using recursion. What recursive backtracking is and why it is an
effective problem-solving technique.
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15.1 Introduction
15.2 Recursion Concepts
15.3 Example Using Recursion: Factorials
15.4 Example Using Recursion: Fibonacci Series
15.5 Recursion and the Method Call Stack
15.6 Recursion vs. Iteration
15.7 Towers of Hanoi
15.8 Fractals
15.9 Recursive Backtracking
15.10 Wrap-Up
15.11 Internet and Web Resources
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15.1 Introduction
• Earlier programs structured as methods that call one another in a disciplined, hierarchical manner
• Recursive methods– Call themselves
– Useful for some problems to define a method to call itself
– Can be called directly or indirectly through another method
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Fig. 15.1 | Summary of the 32 recursion examples and exercises in this text. (Part 1 of 2)
Chapter Recursion examples and exercises in this book
15 Factorial Method (Fig. 15.3 and Fig. 15.4) Fibonacci Method (Fig. 15.5 and Fig. 15.6) Towers of Hanoi (Fig. 15.13 and Fig. 15.14) Fractals (Fig. 15.21 and Fig. 15.22) What Does This Code Do? (Exercise 15.7, Exercise 15.12 and Exercise 15.13) Find the Error in the Following Code (Exercise 15.8) Raising an Integer to an Integer Power (Exercise 15.9) Visualizing Recursion (Exercise 15.10) Greatest Common Divisor (Exercise 15.11) Determine Whether a String Is a Palindrome (Exercise 15.14) Eight Queens (Exercise 15.15) Print an Array (Exercise 15.16) Print an Array Backward (Exercise 15.17) Minimum Value in an Array (Exercise 15.18) Star Fractal (Exercise 15.19) Maze Traversal Using Recursive Backtracking (Exercise 15.20) Generating Mazes Randomly (Exercise 15.21) Mazes of Any Size (Exercise 15.22) Time Needed to Calculate a Fibonacci Number (Exercise 15.23)
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Fig. 15.1 | Summary of the 32 recursion examples and exercises in this text. (Part 2 of 2)
Chapter Recursion examples and exercises in this book
16 Merge Sort (Fig. 16.10 and Fig. 16.11) Linear Search (Exercise 16.8) Binary Search (Exercise 16.9) Quicksort (Exercise 16.10)
17 Binary-Tree Insert (Fig. 17.17) Preorder Traversal of a Binary Tree (Fig. 17.17) Inorder Traversal of a Binary Tree (Fig. 17.17) Postorder Traversal of a Binary Tree (Fig. 17.17) Print a Linked List Backward (Exercise 17.20) Search a Linked List (Exercise 17.21)
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15.2 Recursion Concepts
• Recursive problem-solving elements– Base case
• Recursive method capable of solving only simplest case—the base case• If method is called with base case, method returns result
– If method is called with more complex problem, problem divided into two pieces—a piece the method knows how to do and a piece the method does not know how to do (called recursive call or recursion step)
– Recursive call/recursion step• Must resemble original problem but be slightly simpler or smaller version• Method calls fresh copy of itself to work on smaller problem• Normally includes return statement
• Indirect recursion– Recursive method calls another method that eventually makes call back
to recursive method
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15.3 Example Using Recursion: Factorials
• Factorial of n, or n! is the productn · (n – 1) · (n – 2) · … · 1
With 1! equal to 1 and 0! Defined to be 1.
• Can be solved recursively or iteratively (nonrecursively)
• Recursive solution uses following relationship:n! = n · (n – 1)!
• Infinite recursion – recursive calls are continuously made until memory has been exhausted
– Caused by either omitting base case or writing recursion step that does not converge on base case
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1 // Fig. 15.3: FactorialCalculator.java
2 // Recursive factorial method.
3
4 public class FactorialCalculator
5 {
6 // recursive method factorial
7 public long factorial( long number )
8 {
9 if ( number <= 1 ) // test for base case
10 return 1; // base cases: 0! = 1 and 1! = 1
11 else // recursion step
12 return number * factorial( number - 1 );
13 } // end method factorial
14
15 // output factorials for values 0-10
16 public void displayFactorials()
17 {
18 // calculate the factorials of 0 through 10
19 for ( int counter = 0; counter <= 10; counter++ )
20 System.out.printf( "%d! = %d\n", counter, factorial( counter ) );
21 } // end method displayFactorials
22 } // end class FactorialCalculator
Base case returns 1
Portion method knows how to do
Recursion step breaks problem into two parts: one the method knows how to do, one the method does not
Recursive call: Portion method does not know how to do; smaller version of original problem
Original call to recursive method
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Common Programming Error 15.1
Either omitting the base case or writing the recursion step incorrectly so that it does not converge on the base case can cause a logic error known as infinite recursion, where recursive calls are continuously made until memory has been exhausted. This error is analogous to the problem of an infinite loop in an iterative (nonrecursive) solution.
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1 // Fig. 15.4: FactorialTest.java
2 // Testing the recursive factorial method.
3
4 public class FactorialTest
5 {
6 // calculate factorials of 0-10
7 public static void main( String args[] )
8 {
9 FactorialCalculator factorialCalculator = new FactorialCalculator();
10 factorialCalculator.displayFactorials();
11 } // end main
12 } // end class FactorialTest 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800
Calculate and display factorials
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15.4 Example Using Recursion: Fibonacci Series
• Fibonacci series begins with 0 and 1 and has property that each subsequent Fibonacci number is the sum of previous two Fibonacci numbers.
• Series occurs in nature, ratio of successive Fibonacci numbers converges on golden ratio or golden mean
• Fibonacci series defined recursively as:fibonacci(0) = 0
fibonacci(1) = 1
fibonacci(n) = fibonacci(n – 1) + fibonacci(n – 2)
• Recursive solution for calculating Fibonacci values results in explosion of recursive method calls
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1 // Fig. 15.5: FibonacciCalculator.java
2 // Recursive fibonacci method.
3
4 public class FibonacciCalculator
5 {
6 // recursive declaration of method fibonacci
7 public long fibonacci( long number )
8 {
9 if ( ( number == 0 ) || ( number == 1 ) ) // base cases
10 return number;
11 else // recursion step
12 return fibonacci( number - 1 ) + fibonacci( number - 2 );
13 } // end method fibonacci
14
15 public void displayFibonacci()
16 {
17 for ( int counter = 0; counter <= 10; counter++ )
18 System.out.printf( "Fibonacci of %d is: %d\n", counter,
19 fibonacci( counter ) );
20 } // end method displayFibonacci
21 } // end class FibonacciCalculator
Two base cases
Two recursive calls
Original call to recursive method
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1 // Fig. 15.6: FibonacciTest.java
2 // Testing the recursive fibonacci method.
3
4 public class FibonacciTest
5 {
6 public static void main( String args[] )
7 {
8 FibonacciCalculator fibonacciCalculator = new FibonacciCalculator();
9 fibonacciCalculator.displayFibonacci();
10 } // end main
11 } // end class FibonacciTest Fibonacci of 0 is: 0 Fibonacci of 1 is: 1 Fibonacci of 2 is: 1 Fibonacci of 3 is: 2 Fibonacci of 4 is: 3 Fibonacci of 5 is: 5 Fibonacci of 6 is: 8 Fibonacci of 7 is: 13 Fibonacci of 8 is: 21 Fibonacci of 9 is: 34 Fibonacci of 10 is: 55
Calculate and display Fibonacci values
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Fig. 15.7 | Set of recursive calls for fibonacci( 3 ).
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Avoid Fibonacci-style recursive programs, because they result in an exponential “explosion” of method calls.
Performance Tip 15.1
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15.5 Recursion and the Method Call Stack
• Method call stack used to keep track of method calls and local variables within a method call
• Just as with nonrecursive programming, recursive method calls are placed at the top of the method call stack
• As recursive method calls return, their activation records are popped off the stack and the previous recursive calls continue executing
• Current method executing is always method whose activation record is at top of stack
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Fig. 15.8 | Method calls made within the call fibonacci( 3 ).
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Fig. 15.9 | Method calls on the program execution stack.
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15.6 Recursion vs. Iteration
• Any problem that can be solved recursively can be solved iteratively
• Both iteration and recursion use a control statement– Iteration uses a repetition statement
– Recursion uses a selection statement
• Iteration and recursion both involve a termination test– Iteration terminates when the loop-continuation condition fails
– Recursion terminates when a base case is reached
• Recursion can be expensive in terms of processor time and memory space, but usually provides a more intuitive solution
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1 // Fig. 15.10: FactorialCalculator.java
2 // Iterative factorial method.
3
4 public class FactorialCalculator
5 {
6 // recursive declaration of method factorial
7 public long factorial( long number )
8 {
9 long result = 1;
10
11 // iterative declaration of method factorial
12 for ( long i = number; i >= 1; i-- )
13 result *= i;
14
15 return result;
16 } // end method factorial
17
18 // output factorials for values 0-10
19 public void displayFactorials()
20 {
21 // calculate the factorials of 0 through 10
22 for ( int counter = 0; counter <= 10; counter++ )
23 System.out.printf( "%d! = %d\n", counter, factorial( counter ) );
24 } // end method displayFactorials
25 } // end class FactorialCalculator
Iterative solution uses counter-controlled repetition
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1 // Fig. 15.11: FactorialTest.java
2 // Testing the iterative factorial method.
3
4 public class FactorialTest
5 {
6 // calculate factorials of 0-10
7 public static void main( String args[] )
8 {
9 FactorialCalculator factorialCalculator = new FactorialCalculator();
10 factorialCalculator.displayFactorials();
11 } // end main
12 } // end class FactorialTest 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800
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Software Engineering Observation 15.1
Any problem that can be solved recursively can also be solved iteratively (nonrecursively). A recursive approach is normally preferred over an iterative approach when the recursive approach more naturally mirrors the problem and results in a program that is easier to understand and debug. A recursive approach can often be implemented with fewer lines of code. Another reason to choose a recursive approach is that an iterative one might not be apparent.
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Avoid using recursion in situations requiring high performance. Recursive calls take time and consume additional memory.
Performance Tip 15.2
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Common Programming Error 15.2
Accidentally having a nonrecursive method call itself either directly or indirectly through another method can cause infinite recursion.
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15.7 Towers of Hanoi
• Classic problem – Priests in Far East are attempting to move a stack of disks from one peg to another. One disk must be moved at a time, at no time may a larger disk be placed above a smaller disk
• Recursive solution:– Move n – 1 disks from peg 1 to peg 2, using peg 3 as temporary
holding area– Move the last disk (the largest) from peg 1 to peg 3– Move the n – 1 disks from peg 2 to peg 3, using peg 1 as a
temporary holding area
• Base case: When only one disk needs to be moved – no temporary holding area needed, disk is simply moved
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Fig. 15.12 | Towers of Hanoi for the case with four disks.
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1 // Fig. 15.13: TowersOfHanoi.java
2 // Program solves the towers of Hanoi problem, and
3 // demonstrates recursion.
4
5 public class TowersOfHanoi
6 {
7 int numDisks; // number of disks to move
8
9 public TowersOfHanoi( int disks )
10 {
11 numDisks = disks;
12 } // end TowersOfHanoi constructor
13
14 // recusively move disks through towers
15 public void solveTowers( int disks, int sourcePeg, int destinationPeg,
16 int tempPeg )
17 {
18 // base case -- only one disk to move
19 if ( disks == 1 )
20 {
21 System.out.printf( "\n%d --> %d", sourcePeg, destinationPeg );
22 return;
23 } // end if
24
Base case: Simply display move
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25 // recursion step -- move disk to tempPeg, then to destinationPeg
26 // move ( disks - 1 ) disks from sourcePeg to tempPeg recursively
27 solveTowers( disks - 1, sourcePeg, tempPeg, destinationPeg );
28
29 // move last disk from sourcePeg to destinationPeg
30 System.out.printf( "\n%d --> %d", sourcePeg, destinationPeg );
31
32 // move ( disks - 1 ) disks from tempPeg to destinationPeg
33 solveTowers( disks - 1, tempPeg, destinationPeg, sourcePeg );
34 } // end method solveTowers
35 } // end class TowersOfHanoi
Move n-1 disks from peg 1 to peg 2
Move last disk from peg 1 to peg 3
Move n-1 disks from peg 2 to peg 3Use peg 1 as temporary holding area
Use peg 3 as temporary holding area
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1 // Fig. 15.14: TowersOfHanoiTest.java
2 // Test the solution to the Towers of Hanoi problem.
3
4 public class TowersOfHanoiTest
5 {
6 public static void main( String args[] )
7 {
8 int startPeg = 1; // value 1 used to indicate startPeg in output
9 int endPeg = 3; // value 3 used to indicate endPeg in output
10 int tempPeg = 2; // value 2 used to indicate tempPeg in output
11 int totalDisks = 3; // number of disks
12 TowersOfHanoi towersOfHanoi = new TowersOfHanoi( totalDisks );
13
14 // initial nonrecursive call: move all disks.
15 towersOfHanoi.solveTowers( totalDisks, startPeg, endPeg, tempPeg );
16 } // end main
17 } // end class TowersOfHanoiTest 1 --> 3 1 --> 2 3 --> 2 1 --> 3 2 --> 1 2 --> 3 1 --> 3
Make initial call to recursive method
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15.8 Fractals
• Fractal – a geometric figure that often can be generated from a pattern repeated recursively an infinite number of times
• Pattern applied to each segment of original figure
• Benoit Mandelbrot introduced term “fractal,” along with specifics of how fractals are created and their practical applications
– Help us better understand patterns in nature, the human body and the universe
– Popular art form
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15.8 Fractals
• Self-similar property – fractals have this property in the case that, when subdivided into parts, each resembles a reduced-size copy of the whole
• If part is exact copy of original, fractal is said to be strictly self similar
• Each time pattern is applied, fractal is said to be at new level or depth
• Fractal examples: Koch Curve, Koch Snowflake
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Fig. 15.15 | Koch Curve fractal.
(a) (b)
(c) (d)
(e) (f)
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Fig. 15.17 | Determining points C and D for level 1 of “Lo fractal.”
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Fig. 15.18 | “Lo fractal” at level 1, with C and D points determined for level 2. [Note: The fractal at level 0 is included as a dashed line as a reminder of where the line was located in relation to the
current fractal.]
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Fig. 15.19 | “Lo fractal” at level 2, with dashed lines from level 1 provided.
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1 // Fig. 15.21: Fractal.java
2 // Demonstrates user interface for drawing a fractal.
3 import java.awt.Color;
4 import java.awt.FlowLayout;
5 import java.awt.event.ActionEvent;
6 import java.awt.event.ActionListener;
7 import javax.swing.JFrame;
8 import javax.swing.JButton;
9 import javax.swing.JLabel;
10 import javax.swing.JPanel;
11 import javax.swing.JColorChooser;
12
13 public class Fractal extends JFrame
14 {
15 private final int WIDTH = 400; // define width of GUI
16 private final int HEIGHT = 480; // define height of GUI
17 private final int MIN_LEVEL = 0;
18 private Color color = Color.BLUE;
19
20 private JButton changeColorJButton, increaseLevelJButton,
21 decreaseLevelJButton;
22 private JLabel levelJLabel;
23 private FractalJPanel drawSpace;
24 private JPanel mainJPanel, controlJPanel;
25
26 // set up GUI
27 public Fractal()
28 {
29 super( "Fractal" );
30
42
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31 // set up control panel
32 controlJPanel = new JPanel();
33 controlJPanel.setLayout( new FlowLayout() );
34
35 // set up color button and register listener
36 changeColorJButton = new JButton( "Color" );
37 controlJPanel.add( changeColorJButton );
38 changeColorJButton.addActionListener(
39 new ActionListener() // anonymous inner class
40 {
41 // process changeColorJButton event
42 public void actionPerformed( ActionEvent event )
43 {
44 color = JColorChooser.showDialog(
45 Fractal.this, "Choose a color", color );
46
47 // set default color, if no color is returned
48 if ( color == null )
49 color = Color.BLUE;
50
51 drawSpace.setColor( color );
52 } // end method actionPerformed
53 } // end anonymous inner class
54 ); // end addActionListener
55
43
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56 // set up decrease level button to add to control panel and
57 // register listener
58 decreaseLevelJButton = new JButton( "Decrease Level" );
59 controlJPanel.add( decreaseLevelJButton );
60 decreaseLevelJButton.addActionListener(
61 new ActionListener() // anonymous inner class
62 {
63 // process decreaseLevelJButton event
64 public void actionPerformed( ActionEvent event )
65 {
66 int level = drawSpace.getLevel();
67 level--; // decrease level by one
68
69 // modify level if possible
70 if ( level >= MIN_LEVEL )
71 {
72 levelJLabel.setText( "Level: " + level );
73 drawSpace.setLevel( level );
74 repaint();
75 } // end if
76 } // end method actionPerformed
77 } // end anonymous inner class
78 ); // end addActionListener
79
Retrieve current levelDecrease level
Set new level
Redraw fractal up to new level
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80 // set up increase level button to add to control panel
81 // and register listener
82 increaseLevelJButton = new JButton( "Increase Level" );
83 controlJPanel.add( increaseLevelJButton );
84 increaseLevelJButton.addActionListener(
85 new ActionListener() // anonymous inner class
86 {
87 // process increaseLevelJButton event
88 public void actionPerformed( ActionEvent event )
89 {
90 int level = drawSpace.getLevel();
91 level++; // increase level by one
92
93 // modify level if possible
94 if ( level >= MIN_LEVEL )
95 {
96 levelJLabel.setText( "Level: " + level );
97 drawSpace.setLevel( level );
98 repaint();
99 } // end if
100 } // end method actionPerformed
101 } // end anonymous inner class
102 ); // end addActionListener
103
104 // set up levelJLabel to add to controlJPanel
105 levelJLabel = new JLabel( "Level: 0" );
106 controlJPanel.add( levelJLabel );
107
Retrieve current levelIncrease level
Set new level
Redraw fractal up to new level
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108 drawSpace = new FractalJPanel( 0 );
109
110 // create mainJPanel to contain controlJPanel and drawSpace
111 mainJPanel = new JPanel();
112 mainJPanel.add( controlJPanel );
113 mainJPanel.add( drawSpace );
114
115 add( mainJPanel ); // add JPanel to JFrame
116
117 setSize( WIDTH, HEIGHT ); // set size of JFrame
118 setVisible( true ); // display JFrame
119 } // end Fractal constructor
120
121 public static void main( String args[] )
122 {
123 Fractal demo = new Fractal();
124 demo.setDefaultCloseOperation( JFrame.EXIT_ON_CLOSE );
125 } // end main
126 } // end class Fractal
46
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1 // Fig. 15.22: FractalJPanel.java
2 // FractalJPanel demonstrates recursive drawing of a fractal.
3 import java.awt.Graphics;
4 import java.awt.Color;
5 import java.awt.Dimension;
6 import javax.swing.JPanel;
7
8 public class FractalJPanel extends JPanel
9 {
10 private Color color; // stores color used to draw fractal
11 private int level; // stores current level of fractal
12
13 private final int WIDTH = 400; // defines width of JPanel
14 private final int HEIGHT = 400; // defines height of JPanel
15
16 // set the initial fractal level to the value specified
17 // and set up JPanel specifications
18 public FractalJPanel( int currentLevel )
19 {
20 color = Color.BLUE; // initialize drawing color to blue
21 level = currentLevel; // set initial fractal level
22 setBackground( Color.WHITE );
23 setPreferredSize( new Dimension( WIDTH, HEIGHT ) );
24 } // end FractalJPanel constructor
25
47
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26 // draw fractal recursively
27 public void drawFractal( int level, int xA, int yA, int xB,
28 int yB, Graphics g )
29 {
30 // base case: draw a line connecting two given points
31 if ( level == 0 )
32 g.drawLine( xA, yA, xB, yB );
33 else // recursion step: determine new points, draw next level
34 {
35 // calculate midpoint between (xA, yA) and (xB, yB)
36 int xC = ( xA + xB ) / 2;
37 int yC = ( yA + yB ) / 2;
38
39 // calculate the fourth point (xD, yD) which forms an
40 // isosceles right triangle between (xA, yA) and (xC, yC)
41 // where the right angle is at (xD, yD)
42 int xD = xA + ( xC - xA ) / 2 - ( yC - yA ) / 2;
43 int yD = yA + ( yC - yA ) / 2 + ( xC - xA ) / 2;
44
45 // recursively draw the Fractal
46 drawFractal( level - 1, xD, yD, xA, yA, g );
47 drawFractal( level - 1, xD, yD, xC, yC, g );
48 drawFractal( level - 1, xD, yD, xB, yB, g );
49 } // end else
50 } // end method drawFractal
51
Coordinates of first point for line where fractal is being appliedCoordinates of second point for line
where fractal is being appliedBase case: Simply draw line, pattern is
not applied
Recursion step: Apply fractal patternCalculate midpoint
Calculate point to form right triangle
Apply pattern to three new lines
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52 // start drawing the fractal
53 public void paintComponent( Graphics g )
54 {
55 super.paintComponent( g );
56
57 // draw fractal pattern
58 g.setColor( color );
59 drawFractal( level, 100, 90, 290, 200, g );
60 } // end method paintComponent
61
62 // set the drawing color to c
63 public void setColor( Color c )
64 {
65 color = c;
66 } // end method setColor
67
68 // set the new level of recursion
69 public void setLevel( int currentLevel )
70 {
71 level = currentLevel;
72 } // end method setLevel
73
Make first call to recursive method whenever window is repainted
49
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74 // returns level of recursion
75 public int getLevel()
76 {
77 return level;
78 } // end method getLevel
79 } // end class FractalJPanel
52
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15.9 Recursive Backtracking
• Recursive Backtracking – process of using recursion to return to earlier decision point
• If one set of recursive calls does not result in solution, program backs up to previous decision point and makes different decision, often resulting in another set of recursive calls
• Examples– Maze problem
– Eight-Queens problem