1. 1 Sorting Algorithms (Part I) Sorting Algoritms (Part I) Overview  What is Sorting?  Some Useful Array Handling Methods.  Selection Sort and its Implementation.  Brief Analysis of Selection Sort.  Insertion Sort and its Implementation.  Brief Analysis of Insertion sort.  Preview: Sorting Algorithms (Part II).

2. 2 Sorting Algorithms (Part I) What is Sorting?  The efficiency of data handling can be substantially increased if the data is sorted. Imagine searching for a word in a dictionary in which the words are not sorted!  Sorting is the process of re-arranging a collection of data items according to some key-field.  It is a very common activity in data management. Even when a list is maintained in some order, there is often a need to re-arrange the list in a different order.  Because it takes so much processing time, sorting is a serious topic in computer science, about which many different algorithms have been written.  However, the question of which sorting algorithms is more efficient is an open question as this depends on many factors, including:  the size of the data,  the initial arrangement of the data – random, sorted, partially sorted or in reverse order,  the structure of the data.  We shall undertake detailed analysis of a number of these algorithms in ICS202 using an analysis method called Big-O notation.  For now we shall make a brief analysis on each algorithm we consider based on two criteria - the number of comparisons and the number of data movements involved.  In this lecture we consider two simple algorithms - Selection Sort and Insertion sort.

3. 3 Sorting Algorithms (Part I) Some Useful Array Handling Methods  Swapping operation is common in most of the sorting algorithms we shall consider.  Also to test each algorithm, we need to generate an array of random values and print it before and after sorting.  Thus, we begin by looking at a class called ArrayUtil, which encapsulates these useful operations. import java.util.Random; public class ArrayUtil { public static int[] randomIntArray(int length, int n) { int[] a = new int[length]; Random generator = new Random(); for (int i = 0; i < a.length; i++) a[i] = generator.nextInt(n); return a; } public static void swap(int[] a, int i, int j) { int temp = a[i]; a[i] = a[j]; a[j] = temp; } public static void print(int[] a) { for (int i = 0; i < a.length; i++) System.out.print(a[i] + " "); System.out.println(); }

4. 4 Sorting Algorithms (Part I) Selection Sort: Implementation  This includes the following steps:  1. Find the smallest (or largest) item in the list  2. Swap this item with the item at the beginning of the list  3. Repeat steps 1 and 2 using the remaining items in the list until it remains only one item in the list.

5. 5 Sorting Algorithms (Part I) Selection Sort: Implementation (Cont’d) public class SelectSort { public static int minimumPosition(int[] a, int from) { int minPos = from; for (int i = from + 1; i < a.length; i++) if (a[i] < a[minPos]) minPos = i; return minPos; } public static void sort (int[] a) { for (int n = 0; n < a.length - 1; n++) { int minPos = minimumPosition(a, n); if (minPos != n) ArrayUtil.swap(a, minPos, n); } public class SelectSortTest { public static void main(String[] args) { int[] a = ArrayUtil.randomIntArray(20, 100); ArrayUtil.print(a); SelectSort.sort(a); ArrayUtil.print(a); } Index OriginalRound1R2R3 0 7 111 1 2 222 2 1 774 3 4 447

6. 6 Sorting Algorithms (Part I) Brief Informal Analysis of Selection Sort  Selection Sort has the advantage of simplicity and is adequate for small list of say, few hundred items.  The main loop executes n (number of items) times. Moreover, in each iteration, it has to search the rest of the items to find the minimum – about n comparisons maximum. Thus it needs at most n 2 comparison.  However, the number of data movements is much less - at most one for each iteration, thus giving a maximum of n data movements. In fact if the array is already sorted, no data movement is involved.  However, because of the n 2 comparisons, selection sort is regarded as a quadratic sorting method.  One problem with selection sort is that it is not intelligent to know that the array is already sorted or partially sorted. That is, it performs the same number of comparison regardless of the original order of the array.

7. 7 Sorting Algorithms (Part I) Insertion Sort: Implementation  This method works as follows: 1.Starting with i=1, take the i th element to be the current element 2.Scan the array from i-1 down to 0, shifting each element greater than the current element down by one position. 3.Insert the current element where the shifting in step 2 stops. 4.Repeat steps 1 to 3 for i=2, 3, …, n-1; where n is the size of the array. IndexOriginalR 1R 2R 3 07211 12722 21174 34447 for (n = 1; n < a.length; n++) { current = a[n]; for (j = n; j>0 && current < a[j-1]; j--) a[j] = a[j-1]; a[j] = current; }

8. 8 Sorting Algorithms (Part I) Insertion Sort: Implementation (Cont’d)

9. 9 Sorting Algorithms (Part I) Insertion Sort: Implementation (Cont’d) public class InsertSort { public static void sort(int[] a) { int current, n, j; for(n = 1; n < a.length; n++) { current = a[n]; for(j = n; j>0 && current < a[j-1]; j--) a[j] = a[j-1]; a[j] = current; } // End of outer for loop } // End of sort() method } // End of InsertSort class public class InsertSortTest { public static void main(String[] args) { int[] a = ArrayUtil.randomIntArray(20, 100); ArrayUtil.print(a); InsertSort.sort(a); ArrayUtil.print(a); } // End of main() method } // End of InsertSortTest class

10. 10 Sorting Algorithms (Part I) Brief Informal Analysis of Insertion Sort  Like Selection sort, Insertion sort also has the advantage of simplicity.  Again the main loop executes n-times.  The number of comparison is usually much less than that of Selection sort since the comparison stops once an element greater or equal to the current element is found.  In fact, the best case is when the array is already sorted in which case there would be only one comparison in each iteration, giving a total of n.  However, in the worst case, when data is in reverse order, there could be up to n comparisons in the last iteration. Thus on the average (for random data) we have approximately ½ n 2 comparisons.  The main disadvantage of insertion sort is with regards to number of data movements. In each iteration, the current element has to be placed in its correct place by moving each element greater than the current by one. Thus on the average we have ½ n 2 data movements.  Thus, Insertion sort is also regarded as a quadratic sorting method.

11. 11 Sorting Algorithms (Part I) A Useful Sorting Demo Applet l Check the following URL address: http://blackcat.brynmawr.edu/~spoonen/JavaProject/sorter.html