1. Arrays An array is a data object that can hold multiple objects, all of the same type. We can think of an array as a storage box which has multiple compartments. The compartments are numbered (starting at 0) so that we can indicate which compartment we are referring to.

2. Creating an Array The notation for an an array of a given type is: [] For example, an array of int s is denoted by: int[]

3. Naming Arrays Just like other types, we can associate a name with a data object by assigning it a label – a variable: int[] grades; This declares grades to be of type array of int. But it does not create the array.

4. Creating an array To actually create the array, we must use the new operator, and specify the size of the array: int[] grades = new int[100]; This creates a new data object, which can hold 100 int s (numbered from 0 to 99) and sets the variable grades to refer to that new data object.

5. Using Arrays In mathematics, we have a notion of a vector, which is variable that holds a number of elements, e.g. x = {1, 7, 10, 20, 2, 9} An element is referred to by a subscript, e.g., x 1 would be 1, x 2 would be 7, and so forth. The variable is read “x sub 1” and “x sub 2.”

6. Java Arrays In Java, we do the same thing, but the notation is different: x[i] is still read “x sub i” and means the ith element of the array x. Not that in Java, arrays begin at 0, not 1.

7. Using Subscripts The value inside the square bracket can be a constant, a variable, or an expression, but must be an int. The point is that we can vary the expression in the brackets, so that x[i] can refer to different values, depending on the value of i. for(int i=0; i<100; i++) // read in 100 ints into grades grades[i] = sc.nextInt();

8. Example Arrays are useful for storing lists of data. For example, suppose that we wanted to calculate the standard deviation of a set of test grades. The formula is: standard deviation = where x is the average of x.

9. Example (cont'd) We would have to read in and store the grades, and compute the average first: int[] grades = new int[100]; int total = 0, num = 0; int grade = sc.nextInt(); while(grade >= 0) { total += grade; grades[num++] = grade; grade = sc.nextInt(); } double ave = (double)total/(double)num;

10. Example (cont'd) We would then have a second loop to compute the standard deviation. This loop can be a count- controlled loop: double total2 = 0; for(int i=0; i<num; i++) total2 = (grades[i] – average)* (grades[i] - average); double sd = Math.sqrt(total2/num);

11. Finding the Minimum We could also write a loop to find the minimum grade. (We've already done this once – this is to practice using arrays and loops.) int min = grades[0]; for(int i=1; i<num; i++) if(grades[i] < min) min = grades[i];

12. Sorting Arrays Note that we can use this to sort the array, i.e., put it in order from smallest to largest grade. We could find the smallest grade, and switch it with the first grade, find the next smallest grade, and switch it with the second grade, and so on. Note that it's not enough to find the smallest grade. We must find the position of the smallest grade in the array.

13. Finding the Position of the Min All we need to do is to remember the position of the minimum – originally 0, and then i, if a smaller grade is found. We must access the array to find the minimum grade in the comparison: int minpos = 0; for(int i=1; i<num; i++) if(grades[i] < grades[minpos]; minpos = i;

14. Swapping Once we find the position of the minimum grade, we can swap that grade with the grade at position 0. The code to swap a and b is: int t = a; a = b; b = t;

15. Swap Method We can write a method that swaps two elements of an array: private void swap(int[] a, int i, int j) { int t = a[i]; a[i] = a[j]; a[j] = t; }

16. Arrays are Data Objects It is important to note that arrays are data objects, and array variables are object variables. Hence, array variables contain a reference to the data object, and not the data object themselves. When we use assignment on arrays, the reference is copied, and the two variables will point to the same data object. Similarly, when calling a method, the parameter and object refer to the same data object.

17. Sort (cont'd) After we find the position of the minimum grade, we can use the method to swap it with the element at position 0: int minpos = 0; for(int i=1; i<num; i++) if(grades[i] < grades[minpos]; minpos = i; swap(grades, 0, minpos);

18. Sort (cont'd) This only puts the smallest grade in place. For the other grades, we will need another loop: for(int j=0; j<num; j++) { int minpos = j; for(int i=minpos+1; i<num; i++) if(grades[i] < grades[minpos]; minpos = i; swap(grades, j, minpos); }

19. Selection Sort Note that minpos should start at j, and the inner loop begins at minpos+1 to find the smallest grade. Also, we note that if all but the last grades are in their right place, then the last grade must be also. So, in the outer loop, the test could be j < num-1. This sort is called Selection Sort, because we select the smallest element and put it into the right place.

20. Bubble Sort Another kind of sort “bubbles” the largest element up to the right place, by comparing each element to the one following it and swapping them if they are in the wrong order. Each pass guarantees that the largest will be forced to the end. After num passes (actually num-1 ) all the elements will be in the right places.

21. Bubble Sort (cont'd) Here's the code to move the largest to the end: for(j=0; j<num-1; j++) if(grades[j] > grades[j+1]) swap(grades, j, j+1) Note that the index j only goes up to num-2, since then j+1 will be num-1, the last element in the array.

22. Bubble Sort (cont'd) Again, we must do this num times (actually, num-1 times, since if all the elements but the last are in the right place, the last one must be too). After each pass, the next largest element is in the right place at the end of the array, so we don't have to go as far each time, i.e., the inner loop can end one earlier.

23. Bubble Sort (cont'd) for(int i=0; i<num-1; i++) for(j=0; j<num-1-i; j++) if(grades[j] > grades[j+1]) swap(grades, j, j+1)

24. Improved Bubblesort We can improve upon the na i ve bubblesort by checking if we do any swapping at all. If no swaps are done, then the array is in order and we can stop (after a pass that verifies that no swaps are done)

25. Advanced Bubblesort for(int i = 0; i < num-1; i++) { boolean swapped = false; for(int j = 0; j < num-1-i; j++) { if(grades[j] > grades[j+1]) { swap(grades, j, j+1); swapped = true; } if(!swapped) break; }

26. Insertion Sort Another sort that is commonly used to sort cards is as follows: Suppose that we have a hand of cards that is already in order. We can add a card by comparing the card to the last card in our hand, and if the last card is bigger, slide it to the right a bit, and then look at the next-to-last card. Slide it over if it's bigger. We stop when we come to a card that's smaller, or we run out of cards. We insert the new card in the empty slot.

27. Insertion Sort To insert one card, t, into a hand that is already in order, a[0..i-1], we do: while(i > 0 && a[i-1] > t) { a[i] = a[i-1];// slide one card over i--; // move to the next card } a[i] = t; // insert t Note that we have to check we haven't run out of cards (i > 0) before we compare the last card to t.

28. Insertion Sort (cont'd) To sort the whole hand, we need another loop to insert all the cards: for(int j=1; j<num-1; j++) { int t = a[j], i = j; while(i > 0 && a[i-i] > t) { a[i] = a[i-1]; i--; } a[i] = t; }