1. Java Programming: From Problem Analysis to Program Design, 4e Chapter 13 Recursion

2. Java Programming: From Problem Analysis to Program Design, 4e2 Chapter Objectives Learn about recursive definitions Explore the base case and the general case of a recursive definition Learn about recursive algorithms

3. Java Programming: From Problem Analysis to Program Design, 4e3 Chapter Objectives (continued) Learn about recursive methods Become aware of direct and indirect recursion Explore how to use recursive methods to implement recursive algorithms

4. Java Programming: From Problem Analysis to Program Design, 4e4 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

5. Java Programming: From Problem Analysis to Program Design, 4e5 Recursive Definitions (continued)

6. Java Programming: From Problem Analysis to Program Design, 4e6 Recursive Definitions (continued) 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

7. Java Programming: From Problem Analysis to Program Design, 4e7 Recursive Definitions (continued) 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

8. Java Programming: From Problem Analysis to Program Design, 4e8 Tracing a Recursive Method Recursive method –Logically, you can think of a recursive method having unlimited copies of itself –Every recursive call has its own: Code Set of parameters Set of local variables

9. Java Programming: From Problem Analysis to Program Design, 4e9 Tracing a Recursive Method (continued) After completing a recursive call –Control goes back to the 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

10. Java Programming: From Problem Analysis to Program Design, 4e10 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

11. Java Programming: From Problem Analysis to Program Design, 4e11 Designing Recursive Methods Understand problem requirements Determine limiting conditions Identify base cases

12. Java Programming: From Problem Analysis to Program Design, 4e12 Designing Recursive Methods (continued) Provide direct solution to each base case Identify general case(s) Provide solutions to general cases in terms of smaller versions of general cases

13. Java Programming: From Problem Analysis to Program Design, 4e13 Recursive Factorial Method public static int fact(int num) { if (num = = 0) return 1; else return num * fact(num – 1); }

14. Java Programming: From Problem Analysis to Program Design, 4e14 Recursive Factorial Method (continued)

15. Java Programming: From Problem Analysis to Program Design, 4e15 Largest Value in Array

16. Java Programming: From Problem Analysis to Program Design, 4e16 if the size of the list is 1 the largest element in the list is the only element in the list else to find the largest element in list[a]...list[b] a. find the largest element in list[a + 1]...list[b] and call it max b. compare list[a] and max if ( list[a] >= max ) the largest element in list[a]...list[b] is list[a] else the largest element in list[a]...list[b] is max Largest Value in Array (continued)

17. Java Programming: From Problem Analysis to Program Design, 4e17 public static int largest(int[] list, int lowerIndex, int upperIndex) { int max; if (lowerIndex == upperIndex) return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if (list[lowerIndex] >= max) return list[lowerIndex]; else return max; } Largest Value in Array (continued)

18. Java Programming: From Problem Analysis to Program Design, 4e18 Execution of largest (list, 0, 3)

19. Java Programming: From Problem Analysis to Program Design, 4e19 Execution of largest (list, 0, 3)

20. Java Programming: From Problem Analysis to Program Design, 4e20 Recursive Fibonacci

21. Java Programming: From Problem Analysis to Program Design, 4e21 Recursive Fibonacci (continued) 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); }

22. Java Programming: From Problem Analysis to Program Design, 4e22 Recursive Fibonacci (continued)

23. Java Programming: From Problem Analysis to Program Design, 4e23 Towers of Hanoi Problem with Three Disks

24. Java Programming: From Problem Analysis to Program Design, 4e24 Towers of Hanoi: Three Disk Solution

25. Java Programming: From Problem Analysis to Program Design, 4e25 Towers of Hanoi: Three Disk Solution (continued)

26. Java Programming: From Problem Analysis to Program Design, 4e26 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 needle " + needle1 + " to needle " + needle3 + ". " ); moveDisks(count - 1, needle2, needle3, needle1); }

27. Java Programming: From Problem Analysis to Program Design, 4e27 Recursion or Iteration? Two ways to solve particular problem –Iteration –Recursion Iterative control structures: use looping to repeat a set of statements Tradeoffs between two options –Sometimes recursive solution is easier –Recursive solution is often slower

28. Java Programming: From Problem Analysis to Program Design, 4e28 Programming Example: Decimal to Binary

29. Java Programming: From Problem Analysis to Program Design, 4e29

30. Java Programming: From Problem Analysis to Program Design, 4e30 Sierpinski Gaskets of Various Orders

31. Java Programming: From Problem Analysis to Program Design, 4e31 Programming Example: Sierpinski Gasket Input: nonnegative 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

32. Java Programming: From Problem Analysis to Program Design, 4e32 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); } Programming Example: Sierpinski Gasket (continued)

33. Java Programming: From Problem Analysis to Program Design, 4e33 Programming Example: Sierpinski Gasket (continued)

34. Java Programming: From Problem Analysis to Program Design, 4e34 Chapter Summary Recursive definitions Recursive algorithms Recursive methods Base cases General cases

35. Java Programming: From Problem Analysis to Program Design, 4e35 Chapter Summary (continued) Tracing recursive methods Designing recursive methods Varieties of recursive methods Recursion vs. iteration Various recursive functions explored