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Foundation of Computing Systems
Lecture 17
Sorting Algorithms II
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Sorting by Exchange
•Interchange (exchange) pairs of elements that are outof order until no more such pair exists.
•Bubble sort
•Shell sort
•Quick sort
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Sorting by Exchange: Bubble Sort
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Bubble Sort: Illustration
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Complexity Analysis of Bubble Sort
•Case 1: The input list is already in sorted order
•Number of comparisons
•Number of movements
M (n) = 0
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Complexity Analysis of Bubble Sort
•Case 2: The input list is sorted but in reverse order
•Number of comparisons
•Number of movements
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Complexity Analysis of Bubble Sort
•Case 3: Elements in the input list are in random order
Number of comparisons
Number of movements
Let pj be the probability that the largest element is in the unsorted part is
in j-th (1≤j≤n-i+1) location.
The average number of swaps in the i-th pass is
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Complexity Analysis of Bubble Sort
•Case 3: Elements in the input list are in random order
Number of movements
The average number of swaps in the i-th pass is
The average number of movements
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Bubble Sort: Summary
Case
Comparisons
Movement
Memory
Remark
Case1
Input list is in sortedorder
Case2
Input list is sorted inreversed order
Case3
Input list is in randomorder
Run time, T(n)
Complexity
Remark
Best case
Worst case
Average case
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Bubble Sort: Refinements
•Funnel sort
•Cocktail sort
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Cocktail Sort
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Sorting by Exchange: Shell Sort
•Sorting methods based on comparison
–Comparisons and hence movements of data take place betweenadjacent entries only
–This leads to a number of redundant comparisons and datamovements
–A mechanism should be followed with which the comparisonscan take in long leaps instead of short
•Donald L. Shell (1959)
–Use increments:
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Shell Sort: Illustration
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Shell Sort: Illustration
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Shell Sort: Illustration
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Issues in Shell Sort
•Algorithm to be used to sort subsequences in shell sort
–Straight insertion sort
–Shell sort is better than the insertion sort
•Lower number of passes than n number of passes in insertion sort
•Deciding the values of increments
•Several choices have been made
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Increments in Shell Sort
1: Only two increments h and 1
•h is approximately =
•Time complexity (average) =
2: ht = 2t -1 for 1≤ t ≤ log2n, where n is the size of the inputlist
•Increments: 2t -1, …, 15, 7, 3, 1
•Time complexity : (worst) (average)
O(n5/3)
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Increments in Shell Sort
3: ht = (3t -1)/2 for 1≤ t ≤ l Where l is chosen as thesmallest integer such that , n being the size of theinput list
•Worst case time complexity is O(n3/2 )
Many more……….
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Complexity Analysis of Shell Sort
Case
Comparisons
Movement
Memory
Remark
Case 1
Input list is in sortedorder
Case 2
Input list is sorted inreverse order
Case 3
Input list is in randomorder
Run time, T(n)
Complexity
Remark
Best case
Worst case
Average case
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Sorting by Exchange: Quick Sort
•Divide-and-Conquer
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Divide-and-Conquer in Quick Sort
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Partition Method in Quick Sort
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Partition Method in Quick Sort
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Partition Method: Illustration
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Partition Method: Illustration
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Partition Method: Illustration
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Partition Method: Illustration
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Partition Method: Illustration
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Quick Sort: Illustration
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Complexity Analysis of Quick Sort
•Memory requirement
Size of the stack =
•Number of comparisons
Let, T(n) represents total time to sort n elements and P(n) representsthe time for perform a partition of a list of n elements
T(n) = P(n) +T(nl) +T(nr) with T(1) = T(0) = 0
where, nl = number of elements in the left sub list
nr = number of elements in the right sub list and 0 ≤ nl, nr < n
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Complexity Analysis of Quick Sort
•Case 1: Elements in the list are in ascending order
Number of comparisons
Number of movements
C(n) = n-1 + C(n-1), with C(1) = C(0) = 0
M(n) = 0
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Complexity Analysis of Quick Sort
•Case 2: Elements in the list are in reverse order
Number of comparisons
Number of movements
C(n) = n-1 + C(n-1), with C(1) = C(0) = 0
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Complexity Analysis of Quick Sort
•Case 3: Elements in the list are in random order
Number of comparisons
with C(1) = C(0) = 0
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Complexity Analysis of Quick Sort
•Case 3: Elements in the list are in random order
Number of Movements
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Complexity Analysis: Summary
Case
Comparisons
Movement
Memory
Remark
Case 1
Input list is insorted order
Case 2
Input list issorted inreversed order
Case 3
Input list is inrandom order
Run time, T(n)
Complexity
Remark
Worst case
Worst case
Best/averagecase
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Variations in Quick Sort
•Randomized quick sort
•Random sampling quick sort
•Singleton’s quick sort
•Multi-partition quick sort
•Leftist quick sort
•Lomuto’s quick sort
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Lomuto’s quick sort
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Summary: Sorting by Exchange
Algorithm
Best case
Worst case
Average case
Bubble sort
Shell sort
Quick sort
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Summary: Sorting by Exchange
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Summary: Sorting by Exchange
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Summary: Sorting by Exchange
