1. 1 Arrays: Matrix Renamed Instructor: Mainak Chaudhuri mainakc@cse.iitk.ac.in

2. 2 Arrays Till now we are able to declare and initialize few variables Reality: need to compute on a large amount of data Arrays are data structures that can hold a series of values –Just a new name for matrix –Just like a matrix index, an array uses an index to access a particular value

3. 3 Initializing an array int justAVector[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}; String myFriendsNames[] = {“John”, “Brian”, “Jill”, “Jack”, “Ivan”}; char firstFewDigits[] = {‘0’, ‘1’, ‘2’, ‘3’}; boolean whichOfMyFriendsAreTall[] = {false, false, true, true, false};

4. 4 Finding average class Average { public static void main (String arg[]) { float dailyRainFall[] = {12.3, 13.5, 4.2, 2.4, 1.1, 0, 10.8}; int i; float average = 0; for (i=0; i<7; i++) { average += dailyRainFall[i]; } average /= 7; System.out.println (“Average rain fall: ” + average + “ mm”); }

5. 5 Finding maximum class Maximum { public static void main (String arg[]) { int n = 100; double numbers[] = new double[n]; Initialize (numbers, n); System.out.println (“Maximum: ” + FindMax (numbers, n)); }

6. 6 Finding maximum public static void Initialize (double numbers[], int length) { int i; for (i=0; i<length; i++) { numbers[i] = Math.sin(2*Math.PI/(i+1)) + Math.cos(2*Math.PI/(i+2)); }

7. 7 Finding maximum public static double FindMax (double numbers[], int length) { double max = numbers[0]; int i; for (i=1; i<length; i++) { if (numbers[i] > max) { max = numbers[i]; } return max; } } // end class

8. 8 Finding maximum Want to print the position of the maximum also –Need to return two values from FindMax –Use a 2-element array as return type

9. 9 Finding max and max index class Maximum { public static void main (String arg[]) { int n = 100; double numbers[] = new double[n]; double result[]; Initialize (numbers, n); result = FindMax (numbers, n); System.out.println ("Maximum: " + result[0] + ", Position: " + (int)result[1]); }

10. 10 Finding max and max index public static void Initialize (double numbers[], int length) { int i; for (i=0; i<length; i++) { numbers[i] = Math.sin(2*Math.PI/(i+1)) + Math.cos(2*Math.PI/(i+2)); }

11. 11 Finding max and max index public static double[] FindMax (double numbers[], int length) { double result[] = {numbers[0], 0}; int i; for (i=1; i<length; i++) { if (numbers[i] > result[0]) { result[0] = numbers[i]; result[1] = i; } return result; } } // end class

12. 12 Array layout in memory Recall that every variable requires some space to be stored in memory –Often the compiler is responsible for allocating this space –In other words, every variable has an address (just like you and I have addresses) –The address is often called a reference of a variable in Java –If I try to print the value at this address, I will get the value of the variable How is an array stored in memory?

13. 13 Array layout in memory The array elements are stored contiguously in memory –numbers[0], numbers[1], … –Starting address is numbers (same as the address of numbers[0]), add 8 to get the address of numbers[1] –doubles are 64 bits in size and memory is always byte addressed (one byte is 8 bits) –Putting array names in method arguments is equivalent to passing the arrays by reference: modifications to arrays inside the method are reflected outside the method –Of course, it is possible to pass individual array elements (not by reference, but by value i.e. as private copies)

14. 14 FindMax: pass by reference? class Maximum { public static void main (String arg[]) { int n = 100; double numbers[] = new double[n]; double result[] = new double[2]; Initialize (numbers, n); result[0] = numbers[0]; result[1] = 0; FindMax (numbers, n, result); // by “ref.” System.out.println ("Maximum: " + result[0] + ", Position: " + (int)result[1]); }

15. 15 FindMax: pass by reference? public static void Initialize (double numbers[], int length) { int i; for (i=0; i<length; i++) { numbers[i] = Math.sin(2*Math.PI/(i+1)) + Math.cos(2*Math.PI/(i+2)); }

16. 16 FindMax: pass by reference? public static void FindMax (double numbers[], int length, double r[]) { int i; for (i=1; i<length; i++) { if (numbers[i] > r[0]) { r[0] = numbers[i]; r[1] = i; } } // end class

17. 17 FindMax: pass by value class Maximum { // This example does not work public static void main (String arg[]) { int n = 100; double numbers[] = new double[n]; double result[] = new double[2]; Initialize (numbers, n); result[0] = numbers[0]; result[1] = 0; FindMax (numbers, n, result[0], result[1]); System.out.println ("Maximum: " + result[0] + ", Position: " + (int)result[1]); }

18. 18 FindMax: pass by value public static void Initialize (double numbers[], int length) { int i; for (i=0; i<length; i++) { numbers[i] = Math.sin(2*Math.PI/(i+1)) + Math.cos(2*Math.PI/(i+2)); }

19. 19 FindMax: pass by value public static void FindMax (double numbers[], int length, double max, double position) { int i; for (i=1; i<length; i++) { if (numbers[i] > max) { max = numbers[i]; position = i; } } // end class

20. 20 Pass by what? Java passes arguments always by value –This is confusing, but true –You can think of an array name as a “reference variable” and we pass the value of that reference variable –The first implementation works because the method is written in such a way that it accepts references Not because parameters are passed by references

21. 21 Announcements We have a class on Saturday 0800-0900 in L7 No lab on Saturday

22. 22 Reversing an array class Reverse { public static void main (String arg[]) { int size = 10; int somethingStrange[] = new int[size]; Initialize (somethingStrange, size); PrintArray (somethingStrange, size); Reverse (somethingStrange, size); PrintArray (somethingStrange, size); }

23. 23 Reversing an array public static void Initialize (int array[], int size) { int i; array[0] = 1; for (i=1; i<size; i++) { array[i] = (array[i-1]*3) % 23; }

24. 24 Reversing an array public static void Reverse (int array[], int size) { int head = 0, tail = size-1; while (head < tail) { Swap (array, head, tail); head++; tail--; }

25. 25 Reversing an array public static void Swap (int array[], int i, int j) { int temp; temp = array[i]; array[i] = array[j]; array[j] = temp; }

26. 26 Reversing an array public static void PrintArray (int array[], int size) { int i; for (i=0;i<size;i++) { System.out.println (array[i]); } } // end class

27. 27 Bubble sort class BubbleSort { public static void main (String arg[]) { int size = 20; double array[] = new double[size]; Initialize (array, size); // not shown PrintArray (array, size); // not shown Sort (array, size); PrintArray (array, size); // not shown }

28. 28 Bubble sort public static void Sort (double array[], int size) { int i, j; for (i=1;i<=size-1;i++) { for (j=0;j<=size-2;j++) { // better: j<=size-i-1 CompareAndSwap (array, j, j+1); } // Invariant: maximum of sub-array is // in position size-i }

29. 29 Bubble sort public static void CompareAndSwap (double array[], int p, int q) { double temp; if (array[p] > array[q]) { temp = array[p]; array[p] = array[q]; array[q] = temp; } } // end class (How many comparisons?)

30. 30 Insertion sort public static void Sort (double array[], int size) { int i, j; for (i=1; i<size; i++) { for (j=0; array[i] > array[j]; j++); if (j < i) { Insert (array, i, j); } // Invariant: the first i+1 element // are sorted }

31. 31 Insertion sort public static void Insert (double array[], int oldpos, int newpos) { int i; double candidate = array[oldpos]; for (i=oldpos-1; i>=newpos; i--) { array[i+1] = array[i]; } array[i+1] = candidate; }

32. 32 Insertion sort How many comparisons? –Need to consider worst case –What does the array look like in the worst case? –What is the best case? Just the number of comparisons does not tell you the whole story –How many assignments do you execute?

33. 33 Merging two sorted arrays Suppose we have sorted the first m and the remaining n-m elements of an array separately We need to merge these two sorted halves to get a complete sorted array Assume that everything is sorted in ascending order

34. 34 Merging two sorted arrays public static void Merge (double array[], int start, int m, int n) { // start would be 0 in this case double temp[] = new double[n-start]; int index = 0, index1, index2, i; for (index1=start, index2=m; (index1 < m) && (index2 < n);) { if (array[index1] < array[index2]) { temp[index] = array[index1]; index++; index1++; } else { temp[index] = array[index2]; index++; index2++; } } // continued in next slide

35. 35 Merging two sorted arrays for(;index1<m;index1++,index++) { temp[index] = array[index1]; } for(;index2<n;index2++,index++) { temp[index] = array[index2]; } for (i=start;i<n;i++) { array[i] = temp[i-start]; }

36. 36 Merge sort Recursively sort half of the array separately and merge them class MergeSort { public static void main (String arg[]) { int n = 20; double array[] = new double[n]; Initialize (array, n);// not shown Sort (array, 0, n-1); } // continued in next slide

37. 37 Merge sort public static void Sort (double array[], int start, int end) { if (start < end) { Sort (array, start, start+(end+1- start)/2-1); Sort (array, start+(end+1-start)/2, end); Merge (array, start, start+(end+1- start)/2, end+1); } } // end class

38. 38 Merge sort Running time? Let it be T(n) for an array of size n T(n) = 2T(n/2) + cn for some constant c Solution to this functional equation (or recurrence) is the running time of merge sort T(n) = c’nlog 2 (n) for some constant c’ and large n –I can absorb the base of log in the constant Note that this is the worst case running time of merge sort –Much better than bubble sort and insertion sort which had worst case running time quadratic in n

39. 39 Prime numbers Previously we checked for primality of an integer n by dividing it by all integers up to √n We only need to divide by the primes up to √n Use an array to remember the primes seen so far

40. 40 Prime numbers class PrimeNumbers { // Find all primes up to n public static void main (String arg[]) { int n = 10000; // Assume n > 1 // Use Dusart’s bound (just using n wastes // too much of memory for large n) int maxNumPrimes = (int)((n/Math.log(n))*(1+1.2762/Math.log(n))); int primes[] = new int[maxNumPrimes]; int numPrimesFound = 0; int i; // continued on next slide

41. 41 Prime numbers for (i=2; i<=n; i++) { if (CheckPrimality (i, primes, numPrimesFound)) { primes[numPrimesFound] = i; numPrimesFound++; } PrintPrimes(primes, numPrimesFound); } // end main // continued on next slide

42. 42 Prime numbers public static boolean CheckPrimality (int n, int primes[], int count) { int i; for (i=0; (i<count) && (primes[i] <= Math.sqrt(n)); i++) { if ((n%primes[i])==0) { return false; } return true; } // continued on next slide

43. 43 Prime numbers public static void PrintPrimes (int primes[], int count) { int i, j=0; for (i=0; i<count; i++) { System.out.print (primes[i] + “ ”); j++; if (j==10) { // print 10 primes per line System.out.print (“\n”); j=0; } System.out.print (“\n”); } } // end class