ITI 1120 Lab #9

1. ITI 1120Lab #9 Contributors: Diana Inkpen, Daniel Amyot, Romelia Plesa, Alan Williams

2. Lab 9 Agenda • Recursion • Examples of recursive algorithms • Introduction to matrices

3. Reading arrays from the keyboard • Use the provided ITI1120 class: ITI1120.readIntLine() • Reads an array of integers • When entering the values, be sure to leave at least one space between them. • The result is of type int[] • You should ask for the length of the array that is returned, to find out its size. ITI1120.readDoubleLine() • Similar to readIntLine(), except for double values ITI1120.readCharLine() • Reads an array of character values • Includes ALL characters, including spaces • Result is of type char[] • Reading a String ITI1120.readString()

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

5. Practice problem #1 - 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)

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

7. Practice problem #1 –– cont’d • Translate the algorithm into a Java method

8. Practice problem #1 – solution – cont’d public static boolean checkDigits( char[] a, int n ) { boolean allDigits; if( a[n-1] >= '0' && a[n-1] <= '9' ) { if ( n == 1 ) { allDigits = true; } else { allDigits = checkDigits( a, n-1 ); } } else { allDigits = false; } return allDigits; }

9. Practice problem #2 • Write a recursive algorithm to test if a given array is in sorted order.

10. Practice problem #2 - 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)

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

12. Practice problem #3 • 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.

13. Practice problem #3 - solution GIVENS: N (the size of the array) RESULT: A (the array) HEADER: A  CreateArray(N) BODY: A  MakeNewArray(N) FillArray(A, N – 1) FillArrayis the recursive algorithm

14. Practice problem #3 – 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)

15. Practice problem #3 – solution – cont’d BODY: false N = 0 ? true FillArray(A, N-1)  A[N]  N

16. Practice Problem #4: 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 … • b if 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 Aand Bare integers greater than or equal to 1.

17. What is the base case? FindGCD(A, B) is … • B if 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.

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

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

20. Euclid’s Algorithm in Java public static int findGCD( int a, int b ) { int gcd; if ( a >= b ) { if ( a % b == 0 ) { gcd = b; } else { gcd = findGCD( b, a % b ); } } else { gcd = findGCD( b, a ); } return gcd; }

21. 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

22. Matrices • A matrix is a two dimensional rectangular grid of numbers: • The dimensions of the matrix are the numbers of rows and columns (in the above case: row dimension 3, column dimension 3). • A value within a matrix is referred to by its row and column indices, in that order. • Math: number rows and columns from 1, from upper left corner • In math notation, M1,2 = 2 • For algorithms (and Java), we will use indices starting from 0, to correspond to the array numbering. • In algorithm notation, M[0][1] = 2

23. Matrix element processing • To visit every element of an array, we had to use a loop. • To visit every element of a matrix, we need to use a loop inside a loop: • Outer loop: go through each row of a matrix • Inner loop: go through each column within one row of a matrix. • Recursion could be used in place of each loop

24. Matrix example Write an algorithm that finds the sum of the upper triangle of a square matrix (i.e. the diagonal and up). 0 1 2 3 4 1 4 5 3 2 6 3 6 4 6 M = 4 3 6 7 2 3 4 2 2 4 2 3 8 3 5 How do we know if an element of a square matrix is on or above the main diagonal? 0 1 2 3 4 line_index <= column_index

25. Matrix example – cont’d GIVENS: M (the matrix) N (the number of rows and columns in the matrix) RESULT: Sum (the sum of all elements in the upper triangle) INTERMEDIATES: R (row index in the matrix) C column index in the matrix) HEADER: Sum  CalculateUpperTriangle(M,N)

26. BODY: Matrix example – cont’d SUM  0 R  0 true false R < N ? C  0 C < N ? true false R  C ? false true  SUM  SUM + M[R][C] R  R + 1 C  C + 1