1. Data Structures Using Java1 Chapter 5 Recursion

2. Data Structures Using Java2 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

3. Data Structures Using Java3 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

4. Data Structures Using Java4 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

5. Data Structures Using Java5 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

6. Data Structures Using Java6 Tracing a Recursive Method Recursive method –Has unlimited copies of itself –Every recursive call has its own code own set of parameters own set of local variables

7. Data Structures Using Java7 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

8. Data Structures Using Java8 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

9. Data Structures Using Java9 Designing Recursive Methods Understand problem requirements Determine limiting conditions Identify base cases

10. Data Structures Using Java10 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

11. Data Structures Using Java11 Recursive Factorial Function public static int fact(int num) { if(num == 0) return 1; else return num * fact(num – 1); }

12. Data Structures Using Java12 Recursive Factorial Trace

13. Data Structures Using Java13 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 list[lowerIndex]; else return max; }

14. Data Structures Using Java14 Execution of largest(list, 0, 3)

15. Data Structures Using Java15 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); }

16. Data Structures Using Java16 Execution of rFibonacci(2,3,5)

17. Data Structures Using Java17 Towers of Hanoi Problem with Three Disks

18. Data Structures Using Java18 Towers of Hanoi: Three Disk Solution

19. Data Structures Using Java19 Towers of Hanoi: Three Disk Solution

20. Data Structures Using Java20 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); }

21. Data Structures Using Java21 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); }

22. Data Structures Using Java22 Execution of decToBin(13,2)

23. Data Structures Using Java23 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.

24. Data Structures Using Java24 Sierpinski Gaskets of Various Orders

25. Data Structures Using Java25 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

26. Data Structures Using Java26 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); }

27. Data Structures Using Java27 Programming Example: Sierpinski Gasket Input

28. Data Structures Using Java28 Programming Example: Sierpinski Gasket Input

29. Data Structures Using Java29 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

30. Data Structures Using Java30 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.

31. Data Structures Using Java31 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

32. Data Structures Using Java32 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

33. Data Structures Using Java33 Solution to 8-Queens Puzzle

34. Data Structures Using Java34 4-Queens Puzzle

35. Data Structures Using Java35 4-Queens Tree

36. Data Structures Using Java36 8 X 8 Square Board

37. Data Structures Using Java37 Chapter Summary Recursive Definitions Recursive Algorithms Recursive methods Base cases General cases

38. Data Structures Using Java38 Chapter Summary Tracing recursive methods Designing recursive methods Varieties of recursive methods Recursion vs. Iteration Backtracking N-Queens puzzle