1. Recursion

2. Learn about recursive definitions Explore the base case and the general case of a recursive definition Learn about recursive algorithms Lecture Objectives

3. Learn about recursive methods Become aware of direct and indirect recursion Explore how to use recursive methods to implement recursive algorithms Lecture Objectives (Cont’d)

4. 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 Recursive Definitions

5. Recursive Definitions (Cont’d) 0! = 1 (By Definition!) n! = n x (n – 1) ! If n > 0 3! = 3 x 2! 2! = 2 x 1! 1! = 1 x 0! 0! = 1 (Base Case!) 1! = 1 x 0! = 1 x 1 = 1 2! = 2 x 1! = 2 x 1 = 2 3! = 3 x 2! = 3 x 2 = 6

6. 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 Recursive Definitions (Cont’d)

7. 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 Recursive Definitions (Cont’d)

8. 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 Recursive Definitions (Cont’d)

9. 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

10. 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 Tracing a Recursive Method (Cont’d)

11. Designing Recursive Methods Understand problem requirements Determine limiting conditions Identify base cases

12. Provide direct solution to each base case Identify general case(s) Provide solutions to general cases in terms of smaller versions of general cases Designing Recursive Methods (Cont’d)

13. Recursive Factorial Method public static int fact(int num) { if (num = = 0) return 1; else return num * fact(num – 1); }

14. Recursive Factorial Method (Cont’d) Figure 1: Execution path of fact(4)

15. Largest Value in Array Figure 2: Array with six elements

16. if the size of the list is 1 - the largest element in the list is the only element in the array 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 (Cont’d)

17. 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; }

18. Execution of largest(list, 0, 3) Figure 3: Array with four elements System.out.println(largest(list, 0, 3));

19. Execution of largest(list, 0, 3) (Cont’d) Figure 4: Execution of largest(list, 0, 3)

20. Recursive Fibonacci

21. Recursive Fibonacci (Cont’d) 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. Recursive Fibonacci (Cont’d) Figure 5: Execution of rFibNum(2, 3, 5)

23. Recursion and the Method Call Stack Method call stack used to keep track of method calls and local variables within a method call Just as with nonrecursive programming, recursive method calls are placed at the top of the method call stack As recursive method calls return, their activation records are popped off the stack and the previous recursive calls continue executing Current method executing is always method whose activation record is at top of stack

24. Figure 6: Method calls on the program execution stack Recursion and the Method Call Stack (Cont’d)

25. Towers of Hanoi Problem with Three Disks Figure 7: Tower of Hanoi Problem with three disks

26. Towers of Hanoi: Three Disk Solution Figure 8: Tower of Hanoi Problem with three disks – Solution 1

27. Towers of Hanoi: Three Disk Solution (Cont’d) Figure 9: Tower of Hanoi Problem with three disks – Solution 2

28. 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); }

29. Recursion or Iteration? Two ways to solve particular problem  Iteration  Recursion Both iteration and recursion use a control statement  Iteration uses a repetition statement  Recursion uses a selection statement

30. Iteration and recursion both involve a termination test  Iteration terminates when the loop-continuation condition fails  Recursion terminates when a base case is reached Recursion can be expensive in terms of processor time and memory space, but usually provides a more intuitive solution Recursion or Iteration? (Cont’d)

31. Programming Example: Decimal to Binary public static void decToBin(int num, int base) { if (num > 0) { decToBin(num / base, base); System.out.print(num % base); }

32. Execution of decToBin(13, 2) Figure 10: Execution of decToBin(13, 2)

33. Sierpinski Gaskets of Various Orders Figure 11: Sierpinski Gaskets OPTIONAL!

34. 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 OPTIONAL!

35. Programming Example: Sierpinski Gasket (Cont’d) 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); } OPTIONAL!

36. Programming Example: Sierpinski Gasket (Cont’d) Figure 12: Sierpinski Gaskets: Recursion depth of 3 OPTIONAL!

37. Lecture Summary Recursive definitions Recursive algorithms Recursive methods Base cases General cases

38. Tracing recursive methods Designing recursive methods Varieties of recursive methods Recursion vs. Iteration Various recursive functions explored Lecture Summary (Cont’d)