1. ITI 1120 Lab #8 Contributors: Diana Inkpen, Daniel Amyot, Sylvia Boyd, Amy Felty, Romelia Plesa, Alan Williams

2. Lab 8 Agenda Recursion –Examples of recursive algorithms –Recursive Java methods

3. Recursion Practice problem #1 Write a recursive algorithm for counting the number of digits in a non-negative integer, N

4. Practice problem #1- solution GIVENS: N INTERMEDIATE: RestOfDigits RESULT: Count (the number of digits in N) HEADER:Count  NumberOfDigits(N)

5. Practice problem #1 – solution – cont’d BODY: RestOfDigits = N / 10 RestOfDigits = 0 ? true Count  1 false Count  NumberOfDigits(RestOfDigits) Count  Count + 1

6. Trace for N = 254 Line NRestOfDigitsCount Initial values 254?? (1) RestOfDigits = N / 10 25 (2) RestOfDigits = 0 ? false (3) CALL Count  NumberOfDigits(RestOfDigits) (4) Count  Count + 1

7. Trace, page 2 Line NRestOfDigitsCount Initial values 25?? (1) RestOfDigits = N / 10 2 (2) RestOfDigits = 0 ? false (3) CALL Count  NumberOfDigits(RestOfDigits) (4) Count  Count + 1 Count  NumberOfDigits(RestOfDigits) Count  NumberOfDigits(N) 25

8. Trace, page 3 Line NRestOfDigitsCount Initial values 2?? (1) RestOfDigits = N / 10 0 (2) RestOfDigits = 0 ? true (5) Count  1 1 Count  NumberOfDigits(RestOfDigits) Count  NumberOfDigits(N) 21

9. Trace, page 2 Line NRestOfDigitsCount Initial values 25?? (1) RestOfDigits = N / 10 2 (2) RestOfDigits = 0 ? false (3) CALL Count  NumberOfDigits(RestOfDigits) 1 (4) Count  Count + 1 2 Count  NumberOfDigits(RestOfDigits) Count  NumberOfDigits(N) 252

10. Trace, page 1 Line NRestOfDigitsCount Initial values 254?? (1) RestOfDigits = N / 10 25 (2) RestOfDigits = 0 ? false (3) CALL Count  NumberOfDigits(RestOfDigits) 2 (4) Count  Count + 1 3

11. Implement this algorithm in Java

12. Let’s see how the code runs In your recursive method: Put System.out.println() statements at the following locations (print the actual value of n ): –Immediately after local variable declarations 1: Entering method with n = 254 –Just before recursive method call: 2: Recursive call from n = 254 –Just after recursive method call: 3: Returned from recursion with n = 254 –Just before return statement 4: Returning from method with n = 254, count = 3 –In the base case 5: Base case with n = 2

13. Practice problem #2 Write a recursive algorithm to test if all the characters in positions 0...N-1 of an array, A, of characters are digits.

14. Practice problem #2 - solution GIVENS: A (an array of characters) N(test up to this position in array) RESULT: AllDigits (Boolean, true if all characters in position 0..N are digits) HEADER: AllDigits  CheckDigits(A,N)

15. Practice problem #2 – solution – cont’d BODY: A[N-1] ≥ ′0′ AND A[N-1]  ′9′ ? true N = 1 ? true AllDigits  True false AllDigits  CheckDigits(A, N-1) false AllDigits  False

16. Practice problem #2 –– cont’d Translate the algorithm into a Java method

17. Practice problem #2 – solution – cont’d

18. Practice problem #3 Write a recursive algorithm to test if a given array is in sorted order.

19. Practice problem #3 - solution GIVENS: A(an array of integers) N(the size of the array) RESULT: Sorted(Boolean: true if the array is sorted) HEADER: Sorted  CheckSorted(A,N)

20. Practice problem #3 – solution – cont’d BODY: A[N–2] < A[N–1] ? true N = 1 ? true Sorted  True false Sorted  CheckSorted(A, N-1) false Sorted  False

21. Practice problem #4 Write a recursive algorithm to create an array containing the values 0 to N-1 Hint: –Sometimes you need 2 algorithms: The first is a “starter” algorithm that does some setup actions, and then starts off the recursion by calling the second algorithm The second is a recursive algorithm that does most of the work.

22. Practice problem #4 - solution GIVENS: N(the size of the array) RESULT: A(the array) HEADER: A  CreateArray(N) BODY: A  MakeNewArray(N) FillArray(A, N – 1) FillArray is the recursive algorithm

23. Practice problem #4 – solution – cont’d Algorithm FillArray GIVENS: A(an array) N(the largest position in the array to fill) MODIFIEDS: A RESULT: (none) HEADER: FillArray(A, N)

24. Practice problem #4 – solution – cont’d BODY: N = 0 ? true  false FillArray(A, N-1) A[N]  N

25. Practice Problem #5: Euclid’s algorithm The greatest common divisor (GCD) of two positive integers is the largest integer that divides both values with remainders of 0. Euclid’s algorithm for finding the greatest common divisor is as follows: gcd(a, b) is … bif a ≥ b and a mod b is 0 gcd(b, a) if a < b gcd(b, a mod b)otherwise Write a recursive algorithm that takes two integers A and B and returns their greatest common divisor. You may assume that A and B are integers greater than or equal to 1.

26. What is the base case? FindGCD(A, B) is … Bif A ≥ B and A MOD B is 0 FindGCD(B, A) if A < B FindGCD(B, A MOD B)otherwise Question: will this algorithm always reach the base case? –Note that A MOD B is at most B – 1.

27. Euclid’s Algorithm GIVENS: A, B (Two integers > 0) RESULT: GCD(greatest common divisor of A and B) HEADER: GCD  FindGCD(A, B)

28. Euclid’s Algorithm BODY: A MOD B = 0 ? true GCD  B false GCD  FindGCD(B, A MOD B) A ≥ B ? GCD  FindGCD(B, A) true false

29. Euclid’s Algorithm in Java

30. Test this method Create a class Euclid that includes the method findGCD, and also a main( ) method that will ask the user to enter two values and print their GCD. Try the following: a = 1234, b = 4321 a = 8192, b = 192