Comparison-Based Sorting & Analysis Smt Genap

Outline Several sorting algorithms: ◦ Bubble Sort ◦ Selection Sort ◦ Insertion Sort ◦ Shell Sort For each algorithm: ◦ Basic Idea ◦ Example ◦ Implementation ◦ Algorithm Analysis Smt Genap

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. Smt Genap

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. Smt Genap

Bubble Sort: Example Notice that at least one element will be in the correct position each iteration Smt Genap

Bubble Sort: Example Smt Genap Stop here… why?

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; } Smt Genap

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. Smt Genap

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. Smt Genap

Selection Sort: Example Smt Genap

Selection Sort: Example Smt Genap

Selection Sort: Example Smt Genap

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; }... } Smt Genap

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;... } Smt Genap

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? Smt Genap

Insertion Sort: Idea Idea: sorting cards. ◦ 8 | ◦ 5 8 | ◦ | ◦ | 6 3 ◦ | 3 ◦ | Smt Genap

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. Smt Genap

Insertion Sort: Example Smt Genap

Insertion Sort: Example Smt Genap

Insertion Sort: Example Smt Genap

Insertion Sort: Example Smt Genap

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. Smt Genap

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; } Smt Genap

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. Smt Genap

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 Smt Genap

Shell Sort: Idea Smt Genap Original: 5-sort: Sort items with distance 5 element: Donald Shell (1959): Exchange items that are far apart!

Shell Sort: Example Smt Genap Original: After 5-sort: After 3-sort: After 1-sort:

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. Smt Genap

Shell Sort: Analysis Smt Genap O(N 3/2 )?O(N 5/4 )? O(N 7/6 )? So we have 3 nested loops, but Shell Sort is still better than Insertion Sort! Why?

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. Smt Genap

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’ Smt Genap

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; } Smt Genap

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; } Smt Genap

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 (); } Smt Genap

Other kinds of sort Merge Sort Quick Sort Heap sort. Postman sort / Radix Sort. etc. Smt Genap

Further Reading Weiss book, chapter 7: Sorting Algorithm Smt Genap