Data Structures Using Java1 Chapter 9 Sorting Algorithms.
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Data Structures Using Java 1
Chapter 9
Sorting Algorithms
Data Structures Using Java 2
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
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Selection Sort
Selection Sort Methodology: 1. Find smallest (or equivalently largest)
element in the list2. Move it to the beginning (or end) of the list
by swapping it with element in beginning (or end) position
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class OrderedArrayList
public class OrderedArrayList extends ArrayListClass{
public void selectionSort(); {
//statements}...
};
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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
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Swap Functionprivate void swap(int first, int second){ DataElement temp; temp = list[first]; list[first] = list[second]; list[second] = temp;}//end swap
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Selection Sort Functionpublic void selectionSort(){ int loc, minIndex; for(loc = 0; loc < length; loc++) { minIndex = minLocation(loc, length - 1); swap(loc, minIndex); }}
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Selection Sort Example: Array-Based Lists
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Selection Sort Example: Array-Based Lists
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Selection Sort Example: Array-Based Lists
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Selection Sort Example: Array-Based Lists
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Analysis: Selection Sort
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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
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Listsfor(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]
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Lists
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Insertion Sort: Array-Based Listspublic 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
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Insertion Sort: Linked List-Based List
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Insertion Sort: Linked List-Based List
if(firstOutOfOrder.info is less than first.info) move firstOutOfOrder before firstelse{ 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;}
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Insertion Sort: Linked List-Based List
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Insertion Sort: Linked List-Based List
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Insertion Sort: Linked List-Based List
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Insertion Sort: Linked List-Based List
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Analysis: Insertion Sort
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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] or 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
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Lower Bound on Comparison-Based Sort Algorithms
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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
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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
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Lists
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Quick Sort: Array-Based Listsprivate 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 partition9
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Quick Sort: Array-Based Listsprivate void swap(int first, int second){ DataElement temp; temp = list[first]; list[first] = list[second]; list[second] = temp;}//end swap
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Quick Sort: Array-Based Listsprivate 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
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Quick Sort: Array-Based Lists
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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
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Merge Sort Algorithm
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Divide
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Divide
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Merge
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Merge
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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 inthe 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)
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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)
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Heap Sort: Array-Based Lists
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Heap Sort: Array-Based Lists
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Heap Sort: Array-Based Lists
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Heap Sort: Array-Based Lists
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Heap Sort: Array-Based Lists
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Heap Sort: Array-Based Lists
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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
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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.
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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.
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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 RegionCandidate Name Rgn#1 Rgn#2 Rgn#3 Rgn#4 Total-------------- ----- ----- ----- ----- -----Buddy Balto 0 0 0 272 272Doctor Doc 25 71 156 97 349Ducky Donald 110 158 0 0 268...Winner: ???, Votes Received: ???Total votes polled: ???
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Chapter Summary
• Sorting Algorithms– Selection sort – Insertion sort – Quick sort– Merge sort– heap sort
• Algorithm analysis• Priority queues