Data Structures Using Java1 Chapter 9 Sorting Algorithms

Data Structures Using Java2 Chapter Objectives Learn the various sorting algorithms Explore how to implement the selection, insertion, quick, merge, and heap sorting algorithms Discover how the sorting algorithms discussed in this chapter perform Learn how priority queues are implemented

Data Structures Using Java3 Selection Sort Selection Sort Methodology: 1.Find smallest (or equivalently largest) element in the list 2.Move it to the beginning (or end) of the list by swapping it with element in beginning (or end) position

Data Structures Using Java4 class OrderedArrayList public class OrderedArrayList extends ArrayListClass { public void selectionSort(); { //statements }... };

Data Structures Using Java5 Smallest Element in List Function private int minLocation(int first, int last) { int loc, minIndex; minIndex = first; for(loc = first + 1; loc <= last; loc++) if(list[loc] < list[minIndex]) minIndex = loc; return minIndex; }//end minLocation

Data Structures Using Java6 Swap Function private void swap(int first, int second) { DataElement temp; temp = list[first]; list[first] = list[second]; list[second] = temp; }//end swap

Data Structures Using Java7 Selection Sort Function public void selectionSort() { int loc, minIndex; for(loc = 0; loc < length; loc++) { minIndex = minLocation(loc, length - 1); swap(loc, minIndex); }

Data Structures Using Java8 Selection Sort Example: Array-Based Lists

Data Structures Using Java9 Selection Sort Example: Array-Based Lists

Data Structures Using Java10 Selection Sort Example: Array-Based Lists

Data Structures Using Java11 Selection Sort Example: Array-Based Lists

Data Structures Using Java12 Analysis: Selection Sort

Data Structures Using Java13 Insertion Sort Reduces number of key comparisons made in selection sort Can be applied to both arrays and linked lists (examples follow) Methodology –Find first unsorted element in list –Move it to its proper position

Data Structures Using Java14 Insertion Sort: Array-Based Lists

Data Structures Using Java15 Insertion Sort: Array-Based Lists

Data Structures Using Java16 Insertion Sort: Array-Based Lists

Data Structures Using Java17 Insertion Sort: Array-Based Lists

Data Structures Using Java18 Insertion Sort: Array-Based Lists for(firstOutOfOrder = 1; firstOutOfOrder < length; firstOutOfOrder++) if(list[firstOutOfOrder] is less than list[firstOutOfOrder - 1]) { copy list[firstOutOfOrder] into temp initialize location to firstOutOfOrder do { a. move list[location - 1] one array slot down b. decrement location by 1 to consider the next element of the sorted portion of the array } while(location > 0 && the element in the upper sublist at location - 1 is greater than temp) } copy temp into list[location]

Data Structures Using Java19 Insertion Sort: Array-Based Lists

Data Structures Using Java20 Insertion Sort: Array-Based Lists

Data Structures Using Java21 Insertion Sort: Array-Based Lists

Data Structures Using Java22 Insertion Sort: Array-Based Lists public void insertionSort() { int unsortedIndex, location; DataElement temp; for(unsortedIndex = 1; unsortedIndex < length; unsortedIndex++) if(list[unsortedIndex].compareTo(list[unsortedIndex - 1]) < 0) { temp = list[unsortedIndex]; location = unsortedIndex; do { list[location] = list[location - 1]; location--; }while(location > 0 && list[location - 1].compareTo(temp) > 0); list[location] = temp; } }//end insertionSort

Data Structures Using Java23 Insertion Sort: Linked List-Based List

Data Structures Using Java24 Insertion Sort: Linked List-Based List if(firstOutOfOrder.info is less than first.info) move firstOutOfOrder before first else { set trailCurrent to first set current to the second node in the list //search the list while(current.info is less than firstOutOfOrder.info) { advance trailCurrent; advance current; } if(current is not equal to firstOutOfOrder) { //insert firstOutOfOrder between current and trailCurrent lastInOrder.link = firstOutOfOrder.link; firstOutOfOrder.link = current; trailCurrent.link = firstOutOfOrder; } else //firstOutOfOrder is already at the first place lastInOrder = lastInOrder.link; }

Data Structures Using Java25 Insertion Sort: Linked List-Based List

Data Structures Using Java26 Insertion Sort: Linked List-Based List

Data Structures Using Java27 Insertion Sort: Linked List-Based List

Data Structures Using Java28 Insertion Sort: Linked List-Based List

Data Structures Using Java29 Analysis: Insertion Sort

Data Structures Using Java30 Lower Bound on Comparison- Based Sort Algorithms Trace execution of comparison-based algorithm by using graph called comparison tree Let L be a list of n distinct elements, where n > 0. For any j and k, where 1 = j, k = n, either L[j] L[k] Each comparison of the keys has two outcomes; comparison tree is a binary tree Each comparison is a circle, called a node Node is labeled as j:k, representing comparison of L[j] with L[k] If L[j] < L[k], follow the left branch; otherwise, follow the right branch

Data Structures Using Java31 Lower Bound on Comparison- Based Sort Algorithms

Data Structures Using Java32 Lower Bound on Comparison- Based Sort Algorithms Top node in the figure is the root node Straight line that connects the two nodes is called a branch A sequence of branches from a node, x, to another node, y, is called a path from x to y Rectangle, called a leaf, represents the final ordering of the nodes Theorem: Let L be a list of n distinct elements. Any sorting algorithm that sorts L by comparison of the keys only, in its worst case, makes at least O(n*log2n) key comparisons

Data Structures Using Java33 Quick Sort Recursive algorithm Uses the divide-and-conquer technique to sort a list List is partitioned into two sublists, and the two sublists are then sorted and combined into one list in such a way so that the combined list is sorted

Data Structures Using Java34 Quick Sort: Array-Based Lists

Data Structures Using Java35 Quick Sort: Array-Based Lists

Data Structures Using Java36 Quick Sort: Array-Based Lists

Data Structures Using Java37 Quick Sort: Array-Based Lists

Data Structures Using Java38 Quick Sort: Array-Based Lists

Data Structures Using Java39 Quick Sort: Array-Based Lists

Data Structures Using Java40 Quick Sort: Array-Based Lists private int partition(int first, int last) { DataElement pivot; int index, smallIndex; swap(first, (first + last) / 2); pivot = list[first]; smallIndex = first; for(index = first + 1; index <= last; index++) if(list[index].compareTo(pivot) < 0) { smallIndex++; swap(smallIndex, index); } swap(first, smallIndex); return smallIndex; }//end partition 9

Data Structures Using Java41 Quick Sort: Array-Based Lists private void swap(int first, int second) { DataElement temp; temp = list[first]; list[first] = list[second]; list[second] = temp; }//end swap

Data Structures Using Java42 Quick Sort: Array-Based Lists private void recQuickSort(int first, int last) { int pivotLocation; if(first < last) { pivotLocation = partition(first, last); recQuickSort(first, pivotLocation - 1); recQuickSort(pivotLocation + 1, last); } }//end recQuickSort public void quickSort() { recQuickSort(0, length - 1); }//end quickSort

Data Structures Using Java43 Quick Sort: Array-Based Lists

Data Structures Using Java44 Merge Sort Uses the divide-and-conquer technique to sort a list Merge sort algorithm also partitions the list into two sublists, sorts the sublists, and then combines the sorted sublists into one sorted list

Data Structures Using Java45 Merge Sort Algorithm

Data Structures Using Java46 Divide

Data Structures Using Java47 Divide

Data Structures Using Java48 Merge

Data Structures Using Java49 Merge

Data Structures Using Java50 Analysis of Merge Sort Suppose that L is a list of n elements, where n > 0. Let A(n) denote the number of key comparisons in the average case, and W(n) denote the number of key comparisons in the worst case to sort L. It can be shown that: A(n) = n*log2n – 1.26n = O(n*log2n) W(n) = n*log2n – (n–1) = O(n*log2n)

Data Structures Using Java51 Heap Sort Definition: A heap is a list in which each element contains a key, such that the key in the element at position k in the list is at least as large as the key in the element at position 2k + 1 (if it exists), and 2k + 2 (if it exists)

Data Structures Using Java52 Heap Sort: Array-Based Lists

Data Structures Using Java53 Heap Sort: Array-Based Lists

Data Structures Using Java54 Heap Sort: Array-Based Lists

Data Structures Using Java55 Heap Sort: Array-Based Lists

Data Structures Using Java56 Heap Sort: Array-Based Lists

Data Structures Using Java57 Heap Sort: Array-Based Lists

Data Structures Using Java58 Priority Queues: Insertion Assuming the priority queue is implemented as a heap: 1.Insert the new element in the first available position in the list. (This ensures that the array holding the list is a complete binary tree.) 2.After inserting the new element in the heap, the list may no longer be a heap. So to restore the heap: while (parent of new entry < new entry) swap the parent with the new entry

Data Structures Using Java59 Priority Queues: Remove Assuming the priority queue is implemented as a heap, to remove the first element of the priority queue: 1.Copy the last element of the list into the first array position. 2.Reduce the length of the list by 1. 3.Restore the heap in the list.

Data Structures Using Java60 Programming Example: Election Results The presidential election for the student council of your local university is about to be held. Due to confidentiality, the chair of the election committee wants to computerize the voting. The chair is looking for someone to write a program to analyze the data and report the winner. The university has four major divisions, and each division has several departments. For the election, the four divisions are labeled as region 1, region 2, region 3, and region 4. Each department in each division handles its own voting and directly reports the votes received by each candidate to the election committee.

Data Structures Using Java61 Programming Example: Election Results The voting is reported in the following form: firstName lastName regionNumber numberOfVotes The election committee wants the output in the following tabular form: Election Results Votes By Region Candidate Name Rgn#1 Rgn#2 Rgn#3 Rgn#4 Total Buddy Balto Doctor Doc Ducky Donald Winner: ???, Votes Received: ??? Total votes polled: ???

Data Structures Using Java62 Chapter Summary Sorting Algorithms –Selection sort –Insertion sort –Quick sort –Merge sort –heap sort Algorithm analysis Priority queues