1. Data Structures Using Java
Chapter 5
Recursion
Data Structures Using Java

2. Data Structures Using Java
Chapter Objectives
Learn about recursive definitions
Explore the base case and the general case of a recursive definition
Discover what a recursive algorithm is
Learn about recursive methods
Explore how to use recursive methods to implement recursive algorithms
Learn how recursion implements backtracking
Data Structures Using Java

3. Recursive Definitions
Recursion
Process of solving a problem by reducing it to smaller versions of itself
Recursive definition
Definition in which a problem is expressed in terms of a smaller version of itself
Has one or more base cases
Data Structures Using Java

4. Recursive Definitions
Recursive algorithm
Algorithm that finds the solution to a given problem by reducing the problem to smaller versions of itself
Has one or more base cases
Implemented using recursive methods
Recursive method
Method that calls itself
Base case
Case in recursive definition in which the solution is obtained directly
Stops the recursion
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5. Recursive Definitions
General solution
Breaks problem into smaller versions of itself
General case
Case in recursive definition in which a smaller version of itself is called
Must eventually be reduced to a base case
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6. Tracing a Recursive Method
Has unlimited copies of itself
Every recursive call has
its own code
own set of parameters
own set of local variables
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7. Tracing a Recursive Method
After completing recursive call
Control goes back to calling environment
Recursive call must execute completely before control goes back to previous call
Execution in previous call begins from point immediately following recursive call
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8. Recursive Definitions
Directly recursive: a method that calls itself
Indirectly recursive: a method that calls another method and eventually results in the original method call
Tail recursive method: recursive method in which the last statement executed is the recursive call
Infinite recursion: the case where every recursive call results in another recursive call
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9. Designing Recursive Methods
Understand problem requirements
Determine limiting conditions
Identify base cases
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10. Designing Recursive Methods
Provide direct solution to each base case
Identify general case(s)
Provide solutions to general cases in terms of smaller versions of itself
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11. Recursive Factorial Function
public static int fact(int num) {
if(num == 0)
return 1;
else
return num * fact(num – 1); }
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12. Recursive Factorial Trace
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13. Recursive Implementation: Largest Value in Array
public static int largest(int list[], int lowerIndex, int upperIndex) {
int max;
if(lowerIndex == upperIndex) //the size of the sublist is 1
return list[lowerIndex];
else
max = largest(list, lowerIndex + 1, upperIndex);
if(list[lowerIndex] >= max)
return max; }
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14. Execution of largest(list, 0, 3)
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15. Data Structures Using Java
Recursive Fibonacci
public static int rFibNum(int a, int b, int n) {
if(n == 1)
return a;
else if(n == 2)
return b;
else
return rFibNum(a, b, n - 1) + rFibNum(a, b, n - 2); }
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16. Execution of rFibonacci(2,3,5)
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17. Towers of Hanoi Problem with Three Disks
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18. Towers of Hanoi: Three Disk Solution
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19. Towers of Hanoi: Three Disk Solution
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20. Towers of Hanoi: Recursive Algorithm
public static void moveDisks(int count, int needle1, int needle3, int needle2) {
if(count > 0)
moveDisks(count - 1, needle1, needle2, needle3);
System.out.println("Move disk “ + count + “ from “ +
needle1 + “ to “ + needle3 + ".“);
moveDisks(count - 1, needle2, needle3, needle1); }
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21. Decimal to Binary: Recursive Algorithm
public static void decToBin(int num,
int base) {
if(num > 0)
decToBin(num/base, base);
System.out.println(num % base); }
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22. Execution of decToBin(13,2)
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23. Data Structures Using Java
Sierpinski Gasket
Suppose that you have the triangle ABC.
Determine the midpoints P,Q, and R of the sides AB, AC, and BC, respectively.
Draw the lines PQ,QR, and PR.
This creates three triangles APQ, BPR, and CRQ of similar shape as the triangle ABC.
Process of finding midpoints of sides, then drawing lines through midpoints on triangles APQ, BPR, and CRQ is called a Sierpinski gasket of order or level 0, level 1, level 2, and level 3, respectively.
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24. Sierpinski Gaskets of Various Orders
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25. Programming Example: Sierpinski Gasket
Input: non-negative integer indicating level of Sierpinski gasket
Output: triangle shape displaying a Sierpinski gasket of the given order
Solution includes
Recursive method drawSierpinski
Method to find midpoint of two points
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26. Recursive Algorithm to Draw Sierpinski Gasket
private void drawSierpinski(Graphics g, int lev,
Point p1, Point p2, Point p3) {
Point midP1P2;
Point midP2P3;
Point midP3P1;
if(lev > 0)
g.drawLine(p1.x, p1.y, p2.x, p2.y);
g.drawLine(p2.x, p2.y, p3.x, p3.y);
g.drawLine(p3.x, p3.y, p1.x, p1.y);
midP1P2 = midPoint(p1, p2);
midP2P3 = midPoint(p2, p3);
midP3P1 = midPoint(p3, p1);
drawSierpinski(g, lev - 1, p1, midP1P2, midP3P1);
drawSierpinski(g, lev - 1, p2, midP2P3, midP1P2);
drawSierpinski(g, lev - 1, p3, midP3P1, midP2P3); }
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27. Programming Example: Sierpinski Gasket Input
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28. Programming Example: Sierpinski Gasket Input
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29. Recursion or Iteration?
Two ways to solve particular problem
Iteration
Recursion
Iterative control structures: uses looping to repeat a set of statements
Tradeoffs between two options
Sometimes recursive solution is easier
Recursive solution is often slower
Data Structures Using Java

30. Data Structures Using Java
8-Queens Puzzle
Place 8 queens on a chessboard (8 X 8 square board) so that no two queens can attack each other. For any two queens to be non-attacking, they cannot be in the same row, same column, or same diagonals.
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31. Backtracking Algorithm
Attempts to find solutions to a problem by constructing partial solutions
Makes sure that any partial solution does not violate the problem requirements
Tries to extend partial solution towards completion
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32. Backtracking Algorithm
If it is determined that partial solution would not lead to solution
partial solution would end in dead end
algorithm backs up by removing the most recently added part and then tries other possibilities
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33. Solution to 8-Queens Puzzle
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34. Data Structures Using Java
4-Queens Puzzle
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35. Data Structures Using Java
4-Queens Tree
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36. Data Structures Using Java
8 X 8 Square Board
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37. Data Structures Using Java
Chapter Summary
Recursive Definitions
Recursive Algorithms
Recursive methods
Base cases
General cases
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38. Data Structures Using Java
Chapter Summary
Tracing recursive methods
Designing recursive methods
Varieties of recursive methods
Recursion vs. Iteration
Backtracking
N-Queens puzzle
Data Structures Using Java