1. Recursion -- Introduction
Recursion is a fundamental programming technique that can provide an elegant solution certain kinds of problems
Chapter 12 focuses on:
thinking in a recursive manner
programming in a recursive manner
the correct use of recursion
recursion examples
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

2. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Recursive Thinking
A recursive definition is one which uses the word or concept being defined in the definition itself
A recursive definition can be an appropriate way to express a concept
Before applying recursion to programming, it is best to practice thinking recursively
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

3. Recursive Definitions
Consider the following list of numbers:
24, 88, 40, 37
Such a list can be defined as
A LIST is a: number
or a: number comma LIST
That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST
The concept of a LIST is used to define itself
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

4. Recursive Definitions
The recursive part of the LIST definition is used several times, terminating with the non-recursive part:
number comma LIST
, 88, 40, 37
, 40, 37
, 37
number 37
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

5. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Infinite Recursion
All recursive definitions have to have a non-recursive part
If they didn't, there would be no way to terminate the recursive path
Such a definition would cause infinite recursion
This problem is similar to an infinite loop, but the non-terminating "loop" is part of the definition itself
The non-recursive part is often called the base case
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

6. Recursive Definitions
N!, for any positive integer N, is defined to be the product of all integers between 1 and N inclusive
This definition can be expressed recursively as:
1! = 1
N! = N * (N-1)!
The concept of the factorial is defined in terms of another factorial
Eventually, the base case of 1! is reached
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

7. Recursive Definitions5!
5 * 4!
4 * 3!
3 * 2!
2 * 1! 1
120 24 6 2
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

8. Recursive Programming
A method in Java can invoke itself; if set up that way, it is called a recursive method
The code of a recursive method must be structured to handle both the base case and the recursive case
Each call to the method sets up a new execution environment, with new parameters and local variables
As always, when the method completes, control returns to the method that invoked it (which may be an earlier invocation of itself)
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

9. Recursive Programming
Consider the problem of computing the sum of all the numbers between 1 and any positive integer N
This problem can be recursively defined as:
See Recursive_Sum.java
i = 1 N
N-1
N-2
= N +
= N + (N-1) +
= etc.
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

10. Example: Recursive_Sum.java
class Recursive_Sum {
public static void main (String[] args) {
System.out.println ("The sum of 1 to 3 is " + sum (3));
System.out.println ("The sum of 1 to 6 is " + sum (6));
System.out.println ("The sum of 1 to 10 is " + sum (10));
System.out.println ("The sum of 1 to 15 is " + sum (15));
} // method main
public static int sum (int N) {
int result;
if (N == 1)
result = 1;
else result = N + sum (N-1);
return result;
} // method sum
} // class Recursive_Sum
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

11. Recursive Programming
main
sum
sum(3)
sum(1)
sum(2)
result = 1
result = 3
result = 6
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

12. Recursive Programming
Note that just because we can use recursion to solve a problem, doesn't mean we should
For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand
However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version
You must carefully decide whether recursion is the correct technique for any problem
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

13. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Indirect Recursion
A method invoking itself is considered to be direct recursion
A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again
For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again
This is called indirect recursion, and requires all the same care as direct recursion
It is often more difficult to trace and debug
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

14. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Indirect Recursion m1 m2 m3 m1 m2 m3 m1 m2 m3
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

15. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Using Recursion
Recursion is best served when it is easy to define a smaller subset of the problem in terms of the original
Consider the task of repeatedly displaying a set of images in a mosaic that is reminiscent of looking in two mirrors reflecting each other
The base case is reached when the area for the images shrinks to a certain size
See Repeating_Pictures.java
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

16. Example: Repeating_Pictures
import java.applet.Applet;
import java.awt.*;
public class Repeating_Pictures extends Applet {
private final int SIZE = 300; // Size of applet
private final int STOP = 20; // Smallest picture size
private final int OFFSET = 2; // Picture offset from lines
private Image world;
private Image everest;
private Image goat;
public void init() {
world = getImage (getDocumentBase(), "world.gif");
everest = getImage (getDocumentBase(), "everest.gif");
goat = getImage (getDocumentBase(), "goat.gif");
setSize (SIZE, SIZE);
} // method init
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

17. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
// Draws the entire picture recursively.
public void draw_pictures (int size, Graphics page) {
// Draw lines to create four quadrants
page.drawLine (size/2, 0, size/2, size); // vertical
page.drawLine (0, size/2, size, size/2); // horizontal
// Draw three images in different quadrants
page.drawImage (world, 0+OFFSET, size/2+OFFSET,
size/2-(OFFSET*2), size/2-(OFFSET*2), this);
page.drawImage (everest, size/2+OFFSET, 0+OFFSET,
page.drawImage (goat, size/2+OFFSET, size/2+OFFSET,
// Draw the entire picture again in the first quadrant
if (size > STOP)
draw_pictures (size/2, page);
} // method draw_pictures
// Performs the initial call to the draw_pictures method.
public void paint (Graphics page) {
draw_pictures (getSize().width, page);
} // method paint
} // class Repeating_Pictures
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

18. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Using Recursion
A palindrome is a string of characters that reads the same forward and backward:
radar
able was I ere I saw elba
To determine whether a string is a palindrome, you can examine the two outer characters, and work your way in toward the middle of the string
This solution is easily defined recursively
See Palindromes.java
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

19. Example: Palindromes.java
public class Palindromes {
// Creates a Palindrome_Tester object, and tests several strings.
public static void main (String[] args) {
Palindrome_Tester tester = new Palindrome_Tester();
System.out.println ("radar is a palindrome? "
+ tester.ptest ("radar"));
System.out.println ("abcddcba is a palindrome? "
+ tester.ptest ("abcddcba"));
System.out.println ("able was I ere I saw elba is a palindrome? "
+ tester.ptest ("able was I ere I saw elba"));
System.out.println ("hello is a palindrome? "
+ tester.ptest ("hello"));
System.out.println ("abcxycba is a palindrome? "
+ tester.ptest ("abcxycba"));
} // method main
} // class Palindromes
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

20. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Example: Palindromes
class Palindrome_Tester {
// Uses recursion to perform the palindrome test.
public boolean ptest (String str) {
boolean result = false;
if (str.length() <= 1)
result = true;
else
if (str.charAt (0) == str.charAt (str.length()-1))
result = ptest (str.substring (1, str.length()-1));
return result;
} // method ptest
} // class Palindrome_Tester
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

21. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
Using Recursion
A maze is solved by trial and error -- choosing a direction, following a path, returning to a previous point if the wrong move is made
As such, it is another good candidate for a recursive solution
The base case is an invalid move or one which reaches the final destination
See Maze_Search.java
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

22. Example: Maze_Search.java
public class Maze_Search {
// Creates a new maze, prints its original form, attempts
// to solve it, and prints out its final form.
public static void main (String[] args) {
Maze labyrinth = new Maze();
labyrinth.print_maze();
if (labyrinth.solve(0, 0))
System.out.println ("Maze solved!");
else System.out.println ("No solution.");
} // method main
} // class Maze_Search
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

23. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
class Maze {
int[][] grid = {{1,1,1,0,1,1,0,0,0,1,1,1,1},
{1,0,1,1,1,0,1,1,1,1,0,0,1},
{0,0,0,0,1,0,1,0,1,0,1,0,0},
{1,1,1,0,1,1,1,0,1,0,1,1,1},
{1,0,1,0,0,0,0,1,1,1,0,0,1},
{1,0,1,1,1,1,1,1,0,1,1,1,1},
{1,0,0,0,0,0,0,0,0,0,0,0,0},
{1,1,1,1,1,1,1,1,1,1,1,1,1}};
// Prints the maze grid.
public void print_maze () {
System.out.println();
for (int row=0; row < grid.length; row++) {
for (int column=0; column < grid[row].length; column++)
System.out.print (grid[row][column]); }
} // method print_maze
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

24. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
// Attempts to recursively traverse the maze..
public boolean solve (int row, int column) {
boolean done = false;
if (valid (row, column)) {
grid[row][column] = 3; // cell has been tried
if (row == grid.length-1 && column == grid[0].length-1)
done = true; // maze is solved
else {
done = solve (row+1, column); // down
if (!done) done = solve (row, column+1); // right
if (!done) done = solve (row-1, column); // up
if (!done) done = solve (row, column-1); // left }
if (done) // part of the final path
grid[row][column] = 7;
return done;
} // method solve
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.

25. Copyright 1997 by John Lewis and William Loftus. All rights reserved.
// Determines if a specific location is valid.
private boolean valid (int row, int column) {
boolean result = false;
// check if cell is in the bounds of the matrix
if (row >= 0 && row < grid.length &&
column >= 0 && column < grid[0].length)
// check if cell is not blocked and not previously tried
if (grid[row][column] == 1)
result = true;
return result;
} // method valid
} // class Maze
Chapter 12
Copyright 1997 by John Lewis and William Loftus. All rights reserved.