Chapter 5

1. Chapter 5 Recursion Data Structures Using Java

2. 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 Data Structures Using Java

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 Data Structures Using Java

6. 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 Data Structures Using Java

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 Data Structures Using Java

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 Data Structures Using Java

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

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 Data Structures Using Java

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

12. Recursive Factorial Trace Data Structures Using Java

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 list[lowerIndex]; else return max; } } Data Structures Using Java

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

15. 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); } Data Structures Using Java

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

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

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

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

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); } } Data Structures Using Java

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); } } Data Structures Using Java

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

23. 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. Data Structures Using Java

24. Sierpinski Gaskets of Various Orders Data Structures Using Java

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 Data Structures Using Java

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); } } Data Structures Using Java

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

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

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. 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. Data Structures Using Java

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 Data Structures Using Java

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 Data Structures Using Java

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

34. 4-Queens Puzzle Data Structures Using Java

35. 4-Queens Tree Data Structures Using Java

36. 8 X 8 Square Board Data Structures Using Java

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

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