IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Comparison-Based Sorting & Analysis

2 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Several sorting algorithms: Bubble Sort Selection Sort Insertion Sort Shell Sort For each algorithm: Basic Idea Example Implementation Algorithm Analysis Outline

3 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Sorting Sorting = ordering. Sorted = ordered based on a particular way. Generally, collections of data are presented in a sorted manner. Examples of Sorting: Words in a dictionary are sorted (and case distinctions are ignored). Files in a directory are often listed in sorted order. The index of a book is sorted (and case distinctions are ignored). Many banks provide statements that list checks in increasing order (by check number). In a newspaper, the calendar of events in a schedule is generally sorted by date. Musical compact disks in a record store are generally sorted by recording artist. Why? Imagine finding the phone number of your friend in your mobile phone, but the phone book is not sorted.

4 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Bubble Sort: Idea Idea: bubble in water. Bubble in water moves upward. Why? How? When a bubble moves upward, the water from above will move downward to fill in the space left by the bubble.

5 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Bubble Sort: Example Notice that at least one element will be in the correct position each iteration.

6 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Bubble Sort: Example Stop here… why?

7 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Bubble Sort: Implementation void sort(int a[]){ for (int i = a.length; i>=0; i--) { boolean swapped = false; for (int j = 0; j<i; j++) {... if (a[j] > a[j+1]) { int T = a[j]; a[j] = a[j+1]; a[j+1] = T; swapped = true; }... } if (!swapped) return; }

8 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Bubble Sort: Analysis Running time: Worst case: O(N 2 ) Best case: O(N) -- when? why? Variant: bi-directional bubble sort original bubble sort: only works to one direction bi-directional bubble sort: works back and forth.

9 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Selection Sort: Idea 1. We have two group of items: sorted group, and unsorted group 2. Initially, all items are in the unsorted group. The sorted group is empty. We assume that items in the unsorted group unsorted. We have to keep items in the sorted group sorted. 3. Select the “best” (eg. smallest) item from the unsorted group, then put the “best” item at the end of the sorted group. 4. Repeat the process until the unsorted group becomes empty.

10 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Selection Sort: Example

11 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Selection Sort: Example

12 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Selection Sort: Example

13 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Selection Sort: Implementation void sort(int a[]) throws Exception { for (int i = 0; i < a.length; i++) { int min = i; int j; /* Find the smallest element in the unsorted list */ for (j = i + 1; j < a.length; j++)... } if (a[j] < a[min]) { min = j; }... }

14 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Selection Sort: Implementation /* Swap the smallest unsorted element into the end of the sorted list. */ int T = a[min]; a[min] = a[i]; a[i] = T;... }

15 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Selection Sort: Analysis Running time: Worst case: O(N 2 ) Best case: O(N 2 ) Based on big-oh analysis, is selection sort better than bubble sort? Does the actual running time reflect the analysis?

16 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Idea Idea: sorting cards. 8 | | | | | |

17 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Idea 1. We have two group of items: sorted group, and unsorted group 2. Initially, all items in the unsorted group and the sorted group is empty. We assume that items in the unsorted group unsorted. We have to keep items in the sorted group sorted. 3. Pick any item from, then insert the item at the right position in the sorted group to maintain sorted property. 4. Repeat the process until the unsorted group becomes empty.

18 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Insertion Sort: Example

19 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Insertion Sort: Example

20 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Insertion Sort: Example

21 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Insertion Sort: Example

22 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Implementation Insertion sort to sort an array of integers public static void insertionSort (int[] a) { for (int ii = 1; ii < a.length; ii++) { int jj = ii; while (( jj > 0) && (a[jj] < a[jj - 1])) { int temp = a[jj]; a[jj] = a[jj - 1]; a[jj - 1] = temp; jj--; } Note: value of a[jj] always the same  possibility for improvement of efficiency.

23 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Efficient Implementation A slightly more efficient Insertion sort public static void insertionSort2 (int[] a) { for (int ii = 1; ii < a.length; ii++) { int temp = a[ii]; int jj = ii; while (( jj > 0) && (temp < a[jj - 1])) { a[jj] = a[jj - 1]; jj--; } a[jj] = temp; }

24 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Analysis Running time analysis: Worst case: O(N 2 ) Best case: O(N) Is insertion sort faster than selection sort? Notice the similarity and the difference between insertion sort and selection sort.

25 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 A Lower Bound Bubble Sort, Selection Sort, Insertion Sort all have worst case of O(N 2 ). Turns out, for any algorithm that exchanges adjacent items, this is the best worst case: Ω(N 2 ) In other words, this is a lower bound! See proof in Section 8.3 of Weiss

26 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Original: 5-sort: Sort items with distance 5 element: Shell Sort: Idea Donald Shell (1959): Exchange items that are far apart!

27 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb Original: After 5-sort: After 3-sort: Shell Sort: Example After 1-sort:

28 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Shell Sort: Gap Values Gap: the distance between items being sorted. As we progress, the gap decreases. Shell Sort is also called Diminishing Gap Sort. Shell proposed starting gap of N/2, halving at each step. There are many ways of choosing the next gap.

29 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 O(N 3/2 )?O(N 5/4 )? O(N 7/6 )? Shell Sort: Analysis So we have 3 nested loops, but Shell Sort is still better than Insertion Sort! Why?

30 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Generic Sort So far we have methods to sort integers. What about Strings? Employees? Cookies? A new method for each class? No! In order to be sorted, objects should be comparable (less than, equal, greater than). Solution: use an interface that has a method to compare two objects. Remember: A class that implements an interface inherits the interface (method definitions) = interface inheritance, not implementation inheritance.

31 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 The Comparable Interface In Java, generic aspect of “comparable” is defined in an interface in package java.lang : public interface Comparable { public int compareTo (Object ob); } method compareTo returns: negative integer: the object (this) is smaller than the parameter ‘ob’ 0: the object is equal to the parameter ‘ob’ positive integer: the object (this) is greater than the parameter ‘ob’

32 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Interface: Example public class CircleComparable extends Circle implements Comparable { public CircleComparable (double r) {super (r);} public int compareTo (Object other) { CircleComparable otherCircle = (CircleComparable) other; if (radius < otherCircle.getRadius ()) return -1; else if (radius > otherCircle.getRadius ()) return 1; else return 0; }

33 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Insertion Sort: Generic Sort Generic Insertion sort public static void insertionSort3 (Comparable[] a) { for (int ii = 1; ii < a.length; ii++) { Comparable temp = a[ii]; int jj = ii; while (( jj > 0) && (temp.compareTo (a[jj - 1]) < 0)) { a[jj] = a[jj - 1]; jj--; } a[jj] = temp; }

34 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 import java.util.*; public class SortCircle { public static void main (String args[]) { CircleComparable[] ling = new CircleComparable[20]; Random generator = new Random (); for (int ii = 0; ii < ling.length; ii++) { ling[ii] = new CircleComparable ( 1 + generator.nextInt (100)); System.out.print (ling[ii].getRadius () + " "); } System.out.println (); Sort.insertionSort3 (ling); for (int ii = 0; ii < ling.length; ii++) { System.out.print (ling[ii].getRadius () + " "); } System.out.println (); }

35 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb 2007 Other kinds of sort Merge Sort Quick Sort Heap sort. We will discuss this after tree. Postman sort / Radix Sort. etc.

36 Ruli Manurung (Fasilkom UI)IKI10100: Lecture22 nd Feb /resources/animation/ Weiss book, chapter 8: Sorting Algorithm Further Reading