1. Data Structures Using C++
Chapter 10
Sorting Algorithms
Data Structures Using C++

2. Data Structures Using C++
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
Data Structures Using C++

3. Data Structures Using C++
Selection Sort
Sorts list by
Finding smallest (or equivalently largest) element in the list
Moving it to the beginning (or end) of the list by swapping it with element in beginning (or end) position
Data Structures Using C++

4. Smallest Element in List Function
template<class elemType>
int orderedArrayListType<elemType>::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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5. Data Structures Using C++
Swap Function
template<class elemType>
void orderedArrayListType<elemType>::swap(int first, int second) {
elemType temp;
temp = list[first];
list[first] = list[second];
list[second] = temp;
}//end swap
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6. Selection Sort Function
template<class elemType>
void orderedArrayListType<elemType>::selectionSort() {
int loc, minIndex;
for(loc = 0; loc < length - 1; loc++)
minIndex = minLocation(loc, length - 1);
swap(loc, minIndex); }
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7. Selection Sort Example: Array-Based Lists
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8. Selection Sort Example: Array-Based Lists
Data Structures Using C++

9. Selection Sort Example: Array-Based Lists
Data Structures Using C++

10. Selection Sort Example: Array-Based Lists
Data Structures Using C++

11. Analysis: Selection Sort
By analyzing the number of key comparisons, we see that selection sort is an O(n2) algorithm:
Data Structures Using C++

12. Data Structures Using C++
Insertion Sort
Reduces number of key comparisons made in selection sort
Can be applied to both arrays and linked lists (examples follow)
Sorts list by
Finding first unsorted element in list
Moving it to its proper position
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13. Insertion Sort: Array-Based Lists
Data Structures Using C++

14. Insertion Sort: Array-Based Lists
Data Structures Using C++

15. Insertion Sort: Array-Based Lists
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16. Insertion Sort: Array-Based Lists
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17. 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]
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18. Insertion Sort: Array-Based Lists
Data Structures Using C++

19. Insertion Sort: Array-Based Lists
(after skipping a step)
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20. Insertion Sort: Array-Based Lists
template<class elemType>
void orderedArrayListType<elemType>::insertionSort() {
int firstOutOfOrder, location;
elemType temp;
for(firstOutOfOrder = 1; firstOutOfOrder < length;
firstOutOfOrder++)
if(list[firstOutOfOrder] < list[firstOutOfOrder - 1])
temp = list[firstOutOfOrder];
location = firstOutOfOrder; do
list[location] = list[location - 1];
location--;
} while(location > 0 && list[location - 1] > temp);
list[location] = temp; }
}//end insertionSort
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21. Insertion Sort: Linked List-Based
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22. Insertion Sort: Linked List-Based
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;
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23. Analysis: Insertion Sort
What are worst case complexities of each? Best case?
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24. 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, 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 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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25. Lower Bound on Comparison-Based Sort Algorithms
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26. Lower Bound on Comparison-Based Sort Algorithms
Each leaf represents a different final ordering of the input
n inputs implies n! leaves
Since any initial ordering of inputs is possible, all paths through comparison tree are possible
Worst-case number of comparisons = length of longest path from root to any leaf
Binary tree of height h has at most 2h nodes, so 2h > n!
Thus h > log2 n! > log2 (n/3)n = n log2 (n/3) = Ω(n log n)
Theorem: Let L be a list of n distinct elements. Any sorting algorithm that sorts L by comparison of the keys only, in the worst case, makes at least Ω(n log n) key comparisons
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27. Data Structures Using C++
Quick Sort
Recursive algorithm
Uses “divide-and-conquer”
List L 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
Choose a pivot element to partition the list such that everything in the “lower” list is < pivot and everything in the “upper” list is > pivot
Once lower and upper lists are sorted, L is sorted trivially
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28. Quick Sort: Array-Based Lists
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29. Quick Sort: Array-Based Lists
Start with pivot=0 (first position), smallIndex=0, index=1
Loop index through the list
If L[index] < L[pivot], increment smallIndex and swap L[index] with L[smallIndex]
Swap L[pivot] with L[smallIndex]
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30. Quick Sort: Array-Based Lists
pivot
smallIndex
index
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31. Quick Sort: Array-Based Lists
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32. Quick Sort: Array-Based Lists
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33. Quick Sort: Array-Based Lists
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34. Quick Sort: Array-Based Lists
template<class elemType>
int orderedArrayListType<elemType>::partition(int first, int last) {
elemType pivot;
int index, smallIndex;
swap(first, (first + last)/2);
pivot = list[first];
smallIndex = first;
for(index = first + 1; index <= last; index++)
if(list[index] < pivot)
smallIndex++;
swap(smallIndex, index); }
swap(first, smallIndex);
return smallIndex;
Data Structures Using C++

35. Quick Sort: Array-Based Lists
template<class elemType>
void orderedArrayListType<elemType>::swap(int first,int second) {
elemType temp;
temp = list[first];
list[first] = list[second];
list[second] = temp;
} //end swap
Data Structures Using C++

36. Quick Sort: Array-Based Lists
template<class elemType> void orderedArrayListType<elemType>::recQuickSort(int first, int last) { int pivotLocation; if(first <last) pivotLocation = partition(first, last); recQuickSort(first, pivotLocation - 1); recQuickSort(pivotLocation + 1, last); } } //end recQuickSort
template<class elemType>
void orderedArrayListType<elemType>::quickSort() {
recQuickSort(0, length - 1);
}//end quickSort
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37. Data Structures Using C++
Quick Sort Analysis
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38. Data Structures Using C++
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
Merging two sorted lists into a single sorted list is easy and fast
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39. Data Structures Using C++
Merge Sort Algorithm
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40. Data Structures Using C++
Divide
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41. Data Structures Using C++
Divide
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42. Data Structures Using C++
Merge
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43. Data Structures Using C++
Merge
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44. Data Structures Using C++
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 log2 n – 1.26 n = O(n log n)
W(n) = n log2 n – (n – 1) = O(n log n)
Data Structures Using C++

45. Data Structures Using C++
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)
Can also represent as a binary tree where a node is > than both its children
Heap sort builds a heap out of a list (bottom-up), then pulls (max) items off the top, one at a time
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46. Heap Sort: Array-Based Lists
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47. Heap Sort: Array-Based Lists
Check if heap
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48. Heap Sort: Array-Based Lists
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49. Heap Sort: Array-Based Lists
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50. Heap Sort: Array-Based Lists
Data Structures Using C++

51. Heap Sort: Array-Based Lists
Data Structures Using C++

52. Data Structures Using C++
Heapsort
Start with array A of size n, and “heapify” it
Where is A’s max value now?
Swap max with last element
Return to step 1, using only the first n-1 elements
Data Structures Using C++